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openr1_math_3774 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]P... | The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan C... | <think>
Okay, let me try to figure this problem out. So we have this convex quadrilateral BSCT, and the diagonals meet at the midpoint M of ST. Then lines BT and SC meet at point A, and we're given the lengths of AB, BC, and CA: 91, 98, and 105 respectively. Also, AM is perpendicular to BC. We need to find the positive... |
openr1_math_3775 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1.1. (4 points) When going from the first to the third floor, Petya walks 36 steps. When going from the first floor to his own floor in the same building, Vasya walks 72 steps. On which floor does Vasya live?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is about Petya and Vasya ... | 1.1. (4 points) When going from the first to the third floor, Petya walks 36 steps. When going from the first floor to his own floor in the same building, Vasya walks 72 steps. On which floor does Vasya live? | <think>
Okay, let's see. The problem is about Petya and Vasya climbing stairs in a building. We need to figure out on which floor Vasya lives based on the number of steps they take. Hmm. Let's break it down.
First, Petya goes from the first floor to the third floor and walks 36 steps. So, how many floors does Petya cl... |
openr1_math_3776 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
$1 \cdot 68$ If $a, b$ and $c$ represent integers, the expressions $\sqrt{a+\frac{b}{c}}$ and $a \sqrt{\frac{b}{c}}$ are equal if and only if
(A) $a=b=c=1$.
(B) $a=b$ and $c=a=1$.
(C) $c=\frac{b\left(a^{2}-1\right)}{a}$.
(D) $a=b$ and $c$ is any value.
(E) $a=b$ and $c=a-1$.
(6th American High School M... | $1 \cdot 68$ If $a, b$ and $c$ represent integers, the expressions $\sqrt{a+\frac{b}{c}}$ and $a \sqrt{\frac{b}{c}}$ are equal if and only if
(A) $a=b=c=1$.
(B) $a=b$ and $c=a=1$.
(C) $c=\frac{b\left(a^{2}-1\right)}{a}$.
(D) $a=b$ and $c$ is any value.
(E) $a=b$ and $c=a-1$.
(6th American High School Mathematics Examin... | <think>
Okay, let's try to solve this problem step by step. The question says that if a, b, and c are integers, then the expressions sqrt(a + b/c) and a*sqrt(b/c) are equal if and only if one of the given conditions holds. We have to figure out which option is correct.
First, let me write down the equation given in th... |
openr1_math_3777 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. (10 points) Xiaolin performs the following calculation: $M \div 37$, where $M$ is a natural number, and the result is rounded to six decimal places. Xiaolin gets the result 9.684469, which has the correct integer part, and the digits in the decimal part are correct but in the wrong order. The correc... | 1. (10 points) Xiaolin performs the following calculation: $M \div 37$, where $M$ is a natural number, and the result is rounded to six decimal places. Xiaolin gets the result 9.684469, which has the correct integer part, and the digits in the decimal part are correct but in the wrong order. The correct calculation res... | <think>
Alright, let me try to figure out this problem step by step. So, the problem is about Xiaolin dividing a natural number M by 37, rounding the result to six decimal places, and getting 9.684469. However, it's stated that the integer part is correct, but the decimal digits are correct yet in the wrong order. We n... |
openr1_math_3778 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 4.29 Place $n(n \geqslant 1)$ distinct balls into 4 distinct boxes $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, find the number of different ways to place the balls, $g_{n}$, such that $A_{1}$ contains an odd number of balls, and $A_{2}$ contains an even number of balls.<|im_end|>
<|im_start|>assistant
... | Example 4.29 Place $n(n \geqslant 1)$ distinct balls into 4 distinct boxes $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, find the number of different ways to place the balls, $g_{n}$, such that $A_{1}$ contains an odd number of balls, and $A_{2}$ contains an even number of balls. | <think>
Okay, let's see. The problem is to find the number of ways to place n distinct balls into 4 distinct boxes, A1, A2, A3, A4, with the conditions that A1 has an odd number of balls and A2 has an even number of balls. Hmm. So, we need to count the number of such arrangements, which is denoted by g_n.
First, I re... |
openr1_math_3779 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i]<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to figure out this problem. We need to find all positive integers n > 1 such that for any divisor d ... | Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i] | <think>
Okay, let's try to figure out this problem. We need to find all positive integers n > 1 such that for any divisor d of n, both d² - d + 1 and d² + d + 1 are prime numbers. Hmm, interesting. Let me start by understanding what exactly is being asked here.
First, n must be a composite number since it's greater th... |
openr1_math_3780 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Problem 1. We call a 5-tuple of integers arrangeable if its elements can be labeled $a$, $b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all 2017-tuples of integers $n_{1}, n_{2}, \ldots, n_{2017}$ such that if we place them in a circle in clockwise order, then any 5 -tuple of numbers in c... | Problem 1. We call a 5-tuple of integers arrangeable if its elements can be labeled $a$, $b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all 2017-tuples of integers $n_{1}, n_{2}, \ldots, n_{2017}$ such that if we place them in a circle in clockwise order, then any 5 -tuple of numbers in consecutive positi... | <think>
Okay, so I need to figure out all 2017-tuples of integers arranged in a circle where every five consecutive numbers are "arrangeable." An arrangeable 5-tuple is one where you can label the numbers a, b, c, d, e in some order such that a - b + c - d + e equals 29. Hmm. Let me start by understanding what arrangea... |
openr1_math_3781 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Problem 5. Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$
for all real number $x$ and $y$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find all functions f from the real numbers to the real ... | Problem 5. Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$
for all real number $x$ and $y$. | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy the functional equation:
f(x² + f(y)) = f(f(x)) + f(y²) + 2f(xy)
for all real numbers x and y. Hmm, functional equations can be tricky, but maybe I can start by trying some standard approaches. Let me see...
First,... |
openr1_math_3782 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
12. In the arithmetic sequence $\left\{a_{n}\right\}$, it is known that
$$
a_{20} \simeq \frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c} \text {. }
$$
Then $1992 a c-1811 b c-181 a b=$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to solve this problem involving an arithmetic sequence... | 12. In the arithmetic sequence $\left\{a_{n}\right\}$, it is known that
$$
a_{20} \simeq \frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c} \text {. }
$$
Then $1992 a c-1811 b c-181 a b=$ | <think>
Okay, so I need to solve this problem involving an arithmetic sequence. Let me recall what an arithmetic sequence is. It's a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is a₁ and the common difference is d, then the nth term aₙ is given by aₙ = a₁ ... |
openr1_math_3783 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
$x_{1}-x_{2}=1$
#<|im_end|>
<|im_start|>assistant
<think>
Okay, so I've got this equation here: x₁ minus x₂ equals 1. Hmm, I need to figure out what the problem is asking for. Wait, the user just provided the equation. Maybe the question is to solve for the variables x₁ and x₂? But there's only one eq... | $x_{1}-x_{2}=1$
# | <think>
Okay, so I've got this equation here: x₁ minus x₂ equals 1. Hmm, I need to figure out what the problem is asking for. Wait, the user just provided the equation. Maybe the question is to solve for the variables x₁ and x₂? But there's only one equation and two variables, so that usually means there are infinitely... |
openr1_math_3784 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
6. Evaluate
$$
\prod_{\theta=1}^{89}\left(\tan \theta^{\circ} \cos 1^{\circ}+\sin 1^{\circ}\right)
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out how to evaluate this product from θ = 1 to 89 of (tan θ degrees times cos 1 degree plus sin 1 degree). Hmm, that's quite a mouthfu... | 6. Evaluate
$$
\prod_{\theta=1}^{89}\left(\tan \theta^{\circ} \cos 1^{\circ}+\sin 1^{\circ}\right)
$$ | <think>
Okay, let me try to figure out how to evaluate this product from θ = 1 to 89 of (tan θ degrees times cos 1 degree plus sin 1 degree). Hmm, that's quite a mouthful. Let me write it down again to make sure I have it right:
The product is Π_{θ=1}^{89} [tan θ° cos 1° + sin 1°]
First, I need to recall what a produ... |
openr1_math_3785 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
16. Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books... | 16. Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf... | <think>
Okay, so Ali is arranging his books on shelves. Let me try to figure out how many books are on the bottom shelf. The problem gives some information about how he distributes the books step by step, and we know that the top shelf has three books. The answer choices are A through E, with numbers ranging from 20 to... |
openr1_math_3786 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
For how many one-digit positive integers $k$ is the product $k \cdot 234$ divisible by 12 ?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to figure out for how many one-digit positive integers k the product k * 234 is divisible by 12. Let me start by understanding the problem.
First, one-di... | For how many one-digit positive integers $k$ is the product $k \cdot 234$ divisible by 12 ? | <think>
Okay, so I need to figure out for how many one-digit positive integers k the product k * 234 is divisible by 12. Let me start by understanding the problem.
First, one-digit positive integers are 1 through 9, right? So k can be any of those numbers: 1, 2, 3, 4, 5, 6, 7, 8, or 9. The question is asking which of... |
openr1_math_3787 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
(Infinite Solitaire)(^) You all know the rules of solitaire: there are marbles on a board, and you eliminate marbles by jumping over them with another marble, etc. Now consider the plane and place marbles on all integer points of the negative half-plane. The goal of the game is to place a marble as hig... | (Infinite Solitaire)(^) You all know the rules of solitaire: there are marbles on a board, and you eliminate marbles by jumping over them with another marble, etc. Now consider the plane and place marbles on all integer points of the negative half-plane. The goal of the game is to place a marble as high as possible in ... | <think>
Alright, let's try to figure out this infinite solitaire problem. So, the setup is that we have marbles on all integer points in the negative half-plane. That probably means all points (x, y) where y ≤ 0, right? The goal is to get a marble as high as possible in a finite number of moves. The example shows how t... |
openr1_math_3788 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. Misha calculated the products $1 \times 2, 2 \times 3$, $3 \times 4, \ldots, 2017 \times 2018$. For how many of them is the last digit zero?<|im_end|>
<|im_start|>assistant
<think>
Okay, so Misha is calculating all these products from 1×2 up to 2017×2018, and we need to find out how many of them end... | 3. Misha calculated the products $1 \times 2, 2 \times 3$, $3 \times 4, \ldots, 2017 \times 2018$. For how many of them is the last digit zero? | <think>
Okay, so Misha is calculating all these products from 1×2 up to 2017×2018, and we need to find out how many of them end with a zero. Hmm, ending with a zero means the number is a multiple of 10, right? So, the product has to be divisible by 10. Now, 10 factors into 2 and 5. So, for a product to be divisible by ... |
openr1_math_3789 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
10. There are five cards in a box, marked with $1$, $2$, $3$, $4$, $5$ respectively. Three cards are drawn without replacement, and the probability that the largest one is 4 is ( ).
(A) $\frac{1}{10}$
(B) $\frac{1}{5}$
(C) $\frac{3}{10}$
(D) $\frac{2}{5}$
(E) $\frac{1}{2}$<|im_end|>
<|im_start|>assista... | 10. There are five cards in a box, marked with $1$, $2$, $3$, $4$, $5$ respectively. Three cards are drawn without replacement, and the probability that the largest one is 4 is ( ).
(A) $\frac{1}{10}$
(B) $\frac{1}{5}$
(C) $\frac{3}{10}$
(D) $\frac{2}{5}$
(E) $\frac{1}{2}$ | <think>
Okay, so there's this problem here about probability. Let me read it again: There are five cards numbered 1 to 5 in a box. Three cards are drawn without replacement. We need to find the probability that the largest number among the three drawn is exactly 4. The options are given from (A) to (E). Alright, let's ... |
openr1_math_3790 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need ... | Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$) | <think>
Okay, so I need to figure out the minimum number of lines required on a plane so that they intersect in exactly 200 distinct points. Hmm, let me start by recalling some basics.
First, when you have multiple lines on a plane, the maximum number of intersection points they can form is given by the combination f... |
openr1_math_3791 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
14. B. Color the five sides and five diagonals of the convex pentagon $A B C D E$, such that any two segments sharing a common vertex are of different colors. Find the minimum number of colors needed.<|im_end|>
<|im_start|>assistant
<think>
Alright, let's try to figure out this problem. So, we need to ... | 14. B. Color the five sides and five diagonals of the convex pentagon $A B C D E$, such that any two segments sharing a common vertex are of different colors. Find the minimum number of colors needed. | <think>
Alright, let's try to figure out this problem. So, we need to color the five sides and five diagonals of a convex pentagon ABCDE. The rule is that any two segments (sides or diagonals) that share a common vertex must be colored differently. We have to find the minimum number of colors needed. Hmm, okay.
First,... |
openr1_math_3792 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
(Poland 1995)
Let $n$ be a strictly positive integer. Find the minimum value that the quantity
$$
x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n}
$$
can take, where $x_{1}, \ldots, x_{n}$ are strictly positive real numbers whose sum of reciprocals is exactly $n$.<|im_end|>
<... | (Poland 1995)
Let $n$ be a strictly positive integer. Find the minimum value that the quantity
$$
x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n}
$$
can take, where $x_{1}, \ldots, x_{n}$ are strictly positive real numbers whose sum of reciprocals is exactly $n$. | <think>
Okay, so I need to find the minimum value of the expression \( x_1 + \frac{x_2^2}{2} + \frac{x_3^3}{3} + \ldots + \frac{x_n^n}{n} \), where each \( x_i \) is a strictly positive real number, and the sum of their reciprocals is exactly \( n \). That is, \( \frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n} =... |
openr1_math_3793 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
4. Given arithmetic sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}, T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{3 n+2}{2 n+1}$. Then $\frac{a_{7}}{b_{5}}=$ $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here where... | 4. Given arithmetic sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}, T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{3 n+2}{2 n+1}$. Then $\frac{a_{7}}{b_{5}}=$ $\qquad$ | <think>
Okay, so I have this problem here where there are two arithmetic sequences, {a_n} and {b_n}. The sums of their first n terms are given as S_n and T_n respectively. The ratio of these sums is (3n + 2)/(2n + 1). And I need to find the ratio of a_7 to b_5, which is a_7 divided by b_5. Hmm, let's think about how to... |
openr1_math_3794 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2、The solution set of the equation $16 \sin \pi x \cos \pi x=16 x+\frac{1}{x}$ is<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to solve the equation 16 sin(πx) cos(πx) = 16x + 1/x. Hmm, let's see. First, maybe I can simplify the left side of the equation. There's a trigonometric identity for... | 2、The solution set of the equation $16 \sin \pi x \cos \pi x=16 x+\frac{1}{x}$ is | <think>
Okay, so I need to solve the equation 16 sin(πx) cos(πx) = 16x + 1/x. Hmm, let's see. First, maybe I can simplify the left side of the equation. There's a trigonometric identity for sin(2θ) = 2 sinθ cosθ. So, if I use that identity, 16 sin(πx) cos(πx) would be 8*2 sin(πx) cos(πx) which is 8 sin(2πx). So the equ... |
openr1_math_3795 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
9. Solve the equation $f(x)=0$, given that the function $f(x)$ is even and for any $x, y$ the equality $f(x+y)=f(x)+f(y)+$ $xy+2014$ holds.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to solve this problem. The equation to solve is f(x) = 0, where f is an even function, and for any x and y,... | 9. Solve the equation $f(x)=0$, given that the function $f(x)$ is even and for any $x, y$ the equality $f(x+y)=f(x)+f(y)+$ $xy+2014$ holds. | <think>
Okay, let's try to solve this problem. The equation to solve is f(x) = 0, where f is an even function, and for any x and y, the equality f(x + y) = f(x) + f(y) + xy + 2014 holds. Hmm, interesting. So, first, let me try to understand what kind of function f we're dealing with here.
The functional equation given... |
openr1_math_3796 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
An $8 \mathrm{~cm}$ cube has a $4 \mathrm{~cm}$ square hole cut through its centre, as shown. What is the remaining volume, in $\mathrm{cm}^{3}$ ?
(A) 64
(D) 384
(B) 128
(E) 448
(C) 256
 64
(D) 384
(B) 128
(E) 448
(C) 256
 | <think>
Okay, so I need to find the remaining volume of an 8 cm cube after a 4 cm square hole has been cut through its center. Let me start by visualizing the cube. An 8 cm cube means each edge is 8 cm long. The volume of the original cube would then be 8 cm × 8 cm × 8 cm, right? Let me calculate that first. 8 multipli... |
openr1_math_3797 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. A rectangular photograph with dimensions of 30 cm and 4 dm was enlarged several times to create a rectangular billboard. The area of the billboard is 48 m². What are its length and width?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the length and width of a billboard that was cre... | 3. A rectangular photograph with dimensions of 30 cm and 4 dm was enlarged several times to create a rectangular billboard. The area of the billboard is 48 m². What are its length and width? | <think>
Okay, so I need to find the length and width of a billboard that was created by enlarging a photograph multiple times. The original photograph has dimensions 30 cm and 4 dm, and the billboard's area is 48 m². Hmm, let's start by understanding all the units here because they are mixed. The original dimensions ar... |
openr1_math_3798 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2. (10 points) Calculate: $1+2+4+5+7+8+10+11+13+14+16+17+19+20=$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to calculate the sum of these numbers: 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20. Let me see... First, maybe I should just add them up one by one. But that might take too long... | 2. (10 points) Calculate: $1+2+4+5+7+8+10+11+13+14+16+17+19+20=$ | <think>
Okay, so I need to calculate the sum of these numbers: 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20. Let me see... First, maybe I should just add them up one by one. But that might take too long and I could make a mistake. Hmm, maybe there's a pattern here. Let me look at the sequence again.
Starting from ... |
openr1_math_3799 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
[ Diameter, main properties ] [ Signs and properties of the tangent ]
A circle passes through the midpoints of the hypotenuse $A B$ and the leg $B C$ of a right triangle $A B C$ and is tangent to the leg $A C$. In what ratio does the point of tangency divide the leg $A C$.<|im_end|>
<|im_start|>assis... | [ Diameter, main properties ] [ Signs and properties of the tangent ]
A circle passes through the midpoints of the hypotenuse $A B$ and the leg $B C$ of a right triangle $A B C$ and is tangent to the leg $A C$. In what ratio does the point of tangency divide the leg $A C$. | <think>
Okay, so I have this geometry problem here. Let me try to visualize it first. There's a right triangle ABC, where AB is the hypotenuse and BC is one of the legs. The circle in question passes through the midpoints of AB and BC, and it's also tangent to the other leg AC. The question is asking for the ratio in w... |
openr1_math_3800 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\operatorname{arctan}(2 / 3)} \frac{6+\operatorname{tan} x}{9 \sin ^{2} x+4 \cos ^{2} x} d x
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to compute the definite integral from 0 to arctan(2/3) of (6 + tan x) divided by (... | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\operatorname{arctan}(2 / 3)} \frac{6+\operatorname{tan} x}{9 \sin ^{2} x+4 \cos ^{2} x} d x
$$ | <think>
Okay, so I need to compute the definite integral from 0 to arctan(2/3) of (6 + tan x) divided by (9 sin²x + 4 cos²x) dx. Hmm, let's see. First, I should probably simplify the denominator. The denominator is 9 sin²x + 4 cos²x. Maybe I can factor something out or use a trigonometric identity here. Let me think.
... |
openr1_math_3801 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to solve this problem where we have an equation involving an absolute value, and we need to fi... | Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there? | <think>
Okay, so I need to solve this problem where we have an equation involving an absolute value, and we need to find how many different possible values of k there are such that the equation has three distinct real roots. The equation is (x - 1)|x + 1| = x + k/2020. Hmm, okay. Let me start by understanding the equat... |
openr1_math_3802 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.<|im_end|>
<|im_star... | Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits. | <think>
Okay, so I need to prove that the sequences \( (x_n) \) and \( (y_n) \) converge and find their limits. The sequences are defined by \( x_1 = 1 \), \( y_1 = \sqrt{3} \), and the recursive relations:
\[
\begin{cases}
x_{n+1} y_{n+1} - x_n = 0 \\
x_{n+1}^2 + y_n = 2
\end{cases}
\]
for all \( n = 1, 2, 3, \ldots... |
openr1_math_3803 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. Let's call a number $x$ semi-integer if the number $2x$ is an integer. The semi-integer part of a number $x$ is defined as the greatest semi-integer number not exceeding $x$, and we will denote it as $] x[$. Solve the equation $x^{2} + 2 \cdot ] x[ = 6$. (20 points)<|im_end|>
<|im_start|>assistant
<... | 3. Let's call a number $x$ semi-integer if the number $2x$ is an integer. The semi-integer part of a number $x$ is defined as the greatest semi-integer number not exceeding $x$, and we will denote it as $] x[$. Solve the equation $x^{2} + 2 \cdot ] x[ = 6$. (20 points) | <think>
Alright, so I have this equation to solve: \(x^2 + 2 \cdot ]x[ = 6\). The problem defines the semi-integer part of a number \(x\) as the greatest semi-integer not exceeding \(x\), denoted by \(]x[\). A semi-integer is a number \(x\) such that \(2x\) is an integer.
First, let me make sure I understand what a s... |
openr1_math_3804 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## problem statement
Find the point of intersection of the line and the plane.
$\frac{x+2}{1}=\frac{y-2}{0}=\frac{z+3}{0}$
$2 x-3 y-5 z-7=0$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the point where the given line intersects the plane. Let me see... The line is given by those sy... | ## problem statement
Find the point of intersection of the line and the plane.
$\frac{x+2}{1}=\frac{y-2}{0}=\frac{z+3}{0}$
$2 x-3 y-5 z-7=0$ | <think>
Okay, so I need to find the point where the given line intersects the plane. Let me see... The line is given by those symmetric equations: (x + 2)/1 = (y - 2)/0 = (z + 3)/0. Hmm, the denominators for y and z are zero, which is a bit confusing. Wait, no, actually, in symmetric equations, if the denominator is ze... |
openr1_math_3805 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can ... | Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^... | <think>
Okay, let's see. I need to find the smallest positive integer n such that when I expand (x + 1)^n - 1 and take the coefficients modulo 3, the resulting polynomial is divisible by x² + 1. Hmm. So that means when we divide (x + 1)^n - 1 by x² + 1 modulo 3, there should be no remainder.
First, maybe I should rec... |
openr1_math_3806 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
12. Let the equation $x y=6(x+y)$ have all positive integer solutions $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, then $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$
$\qquad$ .<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this equation: xy = 6(x + y). ... | 12. Let the equation $x y=6(x+y)$ have all positive integer solutions $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, then $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$
$\qquad$ . | <think>
Okay, so I have this equation: xy = 6(x + y). And I need to find all positive integer solutions (x, y) and then sum up all the x's and y's. Hmm, let me think how to approach this.
First, since we're dealing with positive integers, both x and y have to be at least 1. But given the equation, if either x or y is ... |
openr1_math_3807 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that squa... | There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.
Note : We sa... | <think>
Okay, so I need to find the minimum number of pieces required on a 2020x2020 grid such that every square has at least two pieces on the diagonals that pass through it. Each square can hold at most one piece. Hmm, let me try to break this down.
First, let's understand the problem statement again. Each square (c... |
openr1_math_3808 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
10. Two students, A and B, are selecting courses from five subjects. They have exactly one course in common, and A selects more courses than B. The number of ways A and B can select courses to meet the above conditions is $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, so we have two students, A... | 10. Two students, A and B, are selecting courses from five subjects. They have exactly one course in common, and A selects more courses than B. The number of ways A and B can select courses to meet the above conditions is $\qquad$ | <think>
Okay, so we have two students, A and B, who are selecting courses from five subjects. The problem states that they have exactly one course in common, and A selects more courses than B. We need to find the number of ways they can select courses under these conditions. Hmm, let me think step by step.
First, let ... |
openr1_math_3809 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
6. Cut a square with a side of 4 into rectangles, the sum of the perimeters of which is 25.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is to cut a square with a side length of 4 into rectangles such that the sum of the perimeters of these rectangles is 25. Hmm, I need to figur... | 6. Cut a square with a side of 4 into rectangles, the sum of the perimeters of which is 25. | <think>
Okay, let's see. The problem is to cut a square with a side length of 4 into rectangles such that the sum of the perimeters of these rectangles is 25. Hmm, I need to figure out if this is possible and if so, how to do it. Let me start by understanding the problem better.
First, the square has an area of 4x4=16... |
openr1_math_3810 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. Given the sets $U=\{1,2,3,4,5,6,7,8\}, A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}$, the number of subsets of set $U$ that are not subsets of set $A$ and are not subsets of set $B$ is<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the number of subsets of set U that are not subsets of set A an... | 1. Given the sets $U=\{1,2,3,4,5,6,7,8\}, A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}$, the number of subsets of set $U$ that are not subsets of set $A$ and are not subsets of set $B$ is | <think>
Okay, so I need to find the number of subsets of set U that are not subsets of set A and not subsets of set B. Let me make sure I understand the problem first.
We have the universal set U = {1,2,3,4,5,6,7,8}, and two subsets A = {1,2,3,4,5} and B = {4,5,6,7,8}. The question is asking for how many subsets of U... |
openr1_math_3811 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{x^{2}+2 x-3}{x^{2}+4 x-5}\right)^{\frac{1}{2-x}}
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to tackle this limit problem. So, the question is to find the limit as x approaches 1 of [(x² + 2... | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{x^{2}+2 x-3}{x^{2}+4 x-5}\right)^{\frac{1}{2-x}}
$$ | <think>
Okay, let's try to tackle this limit problem. So, the question is to find the limit as x approaches 1 of [(x² + 2x - 3)/(x² + 4x - 5)] raised to the power of 1/(2 - x). Hmm, okay. First, I remember that when dealing with limits involving exponents, sometimes it's helpful to take the natural logarithm to simplif... |
openr1_math_3812 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
11.231 The base of the pyramid $S A B C$ is a right isosceles triangle $A B C$, with the length of the hypotenuse $A B=4 \sqrt{2}$. The lateral edge $S C$ of the pyramid is perpendicular to the plane of the base, and its length is 2. Find the measure of the angle and the distance between the lines, one... | 11.231 The base of the pyramid $S A B C$ is a right isosceles triangle $A B C$, with the length of the hypotenuse $A B=4 \sqrt{2}$. The lateral edge $S C$ of the pyramid is perpendicular to the plane of the base, and its length is 2. Find the measure of the angle and the distance between the lines, one of which passes ... | <think>
Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the pyramid SABC. The base is a right isosceles triangle ABC with hypotenuse AB = 4√2. The lateral edge SC is perpendicular to the base and has a length of 2. We need to find the angle and the distance between two lines: on... |
openr1_math_3813 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
A positive integer $N$ has base-eleven representation $\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}$ and base-eight representation $\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Fi... | A positive integer $N$ has base-eleven representation $\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}$ and base-eight representation $\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such... | <think>
Okay, let's try to tackle this problem step by step. So, we have a number N that is represented in base-eleven as abc, where a, b, c are digits. The same number N is also represented in base-eight as 1bca. We need to find the least such N in base ten.
First, let me make sure I understand the problem correctly.... |
openr1_math_3814 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
João has a deck of 52 cards numbered from 1 to 52. A set of three cards is called lucky if the sum of the digits on each card is the same. What is the minimum number of cards João has to take from the deck, without looking, so that among the cards he takes there are necessarily three cards that form a ... | João has a deck of 52 cards numbered from 1 to 52. A set of three cards is called lucky if the sum of the digits on each card is the same. What is the minimum number of cards João has to take from the deck, without looking, so that among the cards he takes there are necessarily three cards that form a set of lucky card... | <think>
Okay, so João has a deck of 52 cards numbered from 1 to 52. We need to find the minimum number of cards he has to take so that among those cards, there must be at least three that form a "lucky" set. A lucky set is defined as three cards where the sum of the digits on each card is the same.
First, I need to u... |
openr1_math_3815 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to add these three fractions: 2/10, 4/100, and 6/1000. Hmm, the denominators are... | $\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$ | <think>
Okay, let's see. I need to add these three fractions: 2/10, 4/100, and 6/1000. Hmm, the denominators are 10, 100, and 1000. Since they're all powers of 10, maybe converting them to decimals would be easier. Let me try that.
Starting with the first one, 2/10. I know that dividing by 10 moves the decimal point ... |
openr1_math_3816 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
4. In how many ways can 8 people be arranged in a line if Alice and Bob must be next to each other, and Carol must be somewhere behind Dan?<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out how to solve this problem. So, we have 8 people, and we need to arrange them in a line with ... | 4. In how many ways can 8 people be arranged in a line if Alice and Bob must be next to each other, and Carol must be somewhere behind Dan? | <think>
Okay, let me try to figure out how to solve this problem. So, we have 8 people, and we need to arrange them in a line with two conditions: Alice and Bob must be next to each other, and Carol must be somewhere behind Dan. Hmm, let's break this down step by step.
First, dealing with Alice and Bob being next to e... |
openr1_math_3817 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out this problem. So, we have the inequality 4x(x - 5) ≤ 375, and we need to find the largest and smallest val... | Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$. | <think>
Okay, let me try to figure out this problem. So, we have the inequality 4x(x - 5) ≤ 375, and we need to find the largest and smallest values of x that satisfy this inequality. Then, subtract the smallest from the largest to get M - m. Hmm, okay, let's start by understanding the inequality step by step.
First, ... |
openr1_math_3818 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
7.073. $9^{x^{2}-1}-36 \cdot 3^{x^{2}-3}+3=0$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to solve the equation \(9^{x^{2}-1} - 36 \cdot 3^{x^{2}-3} + 3 = 0\). Hmm, exponents with variables in the exponent... Maybe I can rewrite the bases to make them the same or related.
First, ... | 7.073. $9^{x^{2}-1}-36 \cdot 3^{x^{2}-3}+3=0$. | <think>
Okay, let's see. I need to solve the equation \(9^{x^{2}-1} - 36 \cdot 3^{x^{2}-3} + 3 = 0\). Hmm, exponents with variables in the exponent... Maybe I can rewrite the bases to make them the same or related.
First, notice that 9 is a power of 3. Specifically, 9 is \(3^2\). So maybe I can rewrite the 9 term as ... |
openr1_math_3819 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. If $A, B, C$ are the three interior angles of $\triangle ABC$, and $A<B<C .\left(C \neq \frac{\pi}{2}\right)$, then which of the following conclusions is correct?
A. $\sin A<\sin C$
B. $\cot A<\cot C$
C. $\tan A<\tan C$
D. $\cos A<\cos C$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The ... | 3. If $A, B, C$ are the three interior angles of $\triangle ABC$, and $A<B<C .\left(C \neq \frac{\pi}{2}\right)$, then which of the following conclusions is correct?
A. $\sin A<\sin C$
B. $\cot A<\cot C$
C. $\tan A<\tan C$
D. $\cos A<\cos C$ | <think>
Okay, let's see. The problem says that in triangle ABC, the angles A, B, C are such that A is less than B, which is less than C, and C is not equal to π/2. So, we have a triangle where the angles are ordered A < B < C, and since all angles in a triangle add up to π, that means C is the largest angle, right? Bec... |
openr1_math_3820 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the sum of all possible values of b where t... | Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | <think>
Okay, so I need to find the sum of all possible values of b where the two equations f(x) = x³ - 3x + b and g(x) = x² + bx - 3 have a common root. Hmm, let's start by recalling that if there's a common root, let's call it α, then α must satisfy both equations. So, substituting α into both equations, we get:
1. ... |
openr1_math_3821 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. In the afternoon, 5 classes need to be scheduled: Physics, Chemistry, Biology, and two self-study periods. If the first class cannot be Biology and the last class cannot be Physics, then the number of different scheduling methods is ( ) kinds.
(A) 36
(B) 39
(C) 60
(D) 78<|im_end|>
<|im_start|>assist... | 3. In the afternoon, 5 classes need to be scheduled: Physics, Chemistry, Biology, and two self-study periods. If the first class cannot be Biology and the last class cannot be Physics, then the number of different scheduling methods is ( ) kinds.
(A) 36
(B) 39
(C) 60
(D) 78 | <think>
Okay, let's see. The problem is about scheduling 5 classes: Physics, Chemistry, Biology, and two self-study periods. The constraints are that the first class cannot be Biology, and the last class cannot be Physics. We need to figure out how many different scheduling methods there are under these conditions. The... |
openr1_math_3822 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 \qquad \textbf{(E)}\ x^4+1$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here: Let S be equal to (x-1)^4 + 4(x-1... | Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 \qquad \textbf{(E)}\ x^4+1$ | <think>
Okay, so I have this problem here: Let S be equal to (x-1)^4 + 4(x-1)^3 + 6(x-1)^2 + 4(x-1) + 1. Then I need to figure out which of the given options S equals. The options are (A) (x-2)^4, (B) (x-1)^4, (C) x^4, (D) (x+1)^4, and (E) x^4+1. Hmm, interesting. Alright, let's start by understanding what's going on h... |
openr1_math_3823 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
12. Gnomes and the Thunderstorm. In a terrible thunderstorm, n gnomes are climbing up a rope ladder in a chain. If there is a thunderclap, each gnome, independently of the others, may fall with probability $p(0<p<1)$. If a gnome falls, they knock down all the gnomes below them. Find:
a) The probabilit... | 12. Gnomes and the Thunderstorm. In a terrible thunderstorm, n gnomes are climbing up a rope ladder in a chain. If there is a thunderclap, each gnome, independently of the others, may fall with probability $p(0<p<1)$. If a gnome falls, they knock down all the gnomes below them. Find:
a) The probability that exactly $k... | <think>
Okay, so there's this problem about gnomes climbing a rope ladder during a thunderstorm. If a thunderclap happens, each gnome has a probability p of falling, independently. But if a gnome falls, they knock down all the gnomes below them. We need to find the probability that exactly k gnomes fall and the expecte... |
openr1_math_3824 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. Each cell of a $5 \times 5$ table is colored in one of three colors: blue, red, or yellow. In each row of the table, the number of yellow cells is not less than the number of red cells and not less than the number of blue cells, and in each column of the table, the number of red cells is not less th... | 1. Each cell of a $5 \times 5$ table is colored in one of three colors: blue, red, or yellow. In each row of the table, the number of yellow cells is not less than the number of red cells and not less than the number of blue cells, and in each column of the table, the number of red cells is not less than the number of ... | <think>
Okay, let's try to tackle this problem. So, we have a 5x5 grid, each cell colored blue, red, or yellow. The conditions are:
1. In each row, the number of yellow cells is not less than red or blue. So, in every row, yellow >= red and yellow >= blue.
2. In each column, the number of red cells is not less than ye... |
openr1_math_3825 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
[ Fermat's Little Theorem ]
Find the remainder when $3^{102}$ is divided by 101.
#<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the remainder when 3^102 is divided by 101. Hmm, the problem mentions Fermat's Little Theorem, so I should probably use that. Let me recall what Fermat's ... | [ Fermat's Little Theorem ]
Find the remainder when $3^{102}$ is divided by 101.
# | <think>
Okay, so I need to find the remainder when 3^102 is divided by 101. Hmm, the problem mentions Fermat's Little Theorem, so I should probably use that. Let me recall what Fermat's Little Theorem states. If I remember correctly, it says that if p is a prime number and a is an integer not divisible by p, then a^(p-... |
openr1_math_3826 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
12. The system of equations $\left\{\begin{array}{l}x+3 y=3, \\ || x|-| y||=1\end{array}\right.$ has ( ) solutions.
(A) 1
(B) 2
(C) 3
(D) 4
(E) 8<|im_end|>
<|im_start|>assistant
<think>
Alright, so I have this system of equations to solve:
First equation: x + 3y = 3
Second equation: ||x| - |y|| = 1
... | 12. The system of equations $\left\{\begin{array}{l}x+3 y=3, \\ || x|-| y||=1\end{array}\right.$ has ( ) solutions.
(A) 1
(B) 2
(C) 3
(D) 4
(E) 8 | <think>
Alright, so I have this system of equations to solve:
First equation: x + 3y = 3
Second equation: ||x| - |y|| = 1
And I need to figure out how many solutions there are. The options are from 1 to 8, but most likely not all of them. Let's take it step by step.
First, the linear equation is x + 3y = 3. That's ... |
openr1_math_3827 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. Let $p$ be a prime number greater than 3, and $p+8$ is also a prime number. Let $S_{1}=p(p+8)$, and $S_{i+1} (i=1,2, \cdots, k-1)$ denote the sum of the digits of $S_{i}$. If $S_{k}$ is a single-digit number, for all possible prime numbers $p$, $S_{k}$ has ( ) different values.
(A) 1
(B) 2
(C) 3
(D)... | 1. Let $p$ be a prime number greater than 3, and $p+8$ is also a prime number. Let $S_{1}=p(p+8)$, and $S_{i+1} (i=1,2, \cdots, k-1)$ denote the sum of the digits of $S_{i}$. If $S_{k}$ is a single-digit number, for all possible prime numbers $p$, $S_{k}$ has ( ) different values.
(A) 1
(B) 2
(C) 3
(D) 4 | <think>
Okay, let's try to figure out this problem step by step. So, we have a prime number p greater than 3, and p + 8 is also a prime. Then S₁ is defined as p multiplied by (p + 8). Then for each subsequent Sᵢ₊₁, it's the sum of the digits of Sᵢ, and this continues until we get a single-digit number Sₖ. The question ... |
openr1_math_3828 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 4. As shown in the figure, points $A, B, C, D$ lie on the same circle, and $BC=DC=4, AE=6$. The lengths of segments $BE$ and $DE$ are both positive integers. What is the length of $BD$? (1988
National Junior High School Mathematics Competition)<|im_end|>
<|im_start|>assistant
<think>
Okay, so I... | Example 4. As shown in the figure, points $A, B, C, D$ lie on the same circle, and $BC=DC=4, AE=6$. The lengths of segments $BE$ and $DE$ are both positive integers. What is the length of $BD$? (1988
National Junior High School Mathematics Competition) | <think>
Okay, so I need to solve this geometry problem where points A, B, C, D are on a circle. The given information is that BC and DC are both 4 units long, AE is 6 units, and both BE and DE are positive integers. The goal is to find the length of BD. Hmm, let me try to visualize the problem first. Since all points l... |
openr1_math_3829 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2. For any real numbers $m$, $n$, $p$, it is required:
(1) to write the corresponding quadratic equation;
(2) to make the equation have $n^{2}-4 m p$ as the discriminant.
Among the following answers:
$$
\begin{array}{l}
m x^{2}+n x+p=0, \\
p x^{2}+n x+m=0, \\
x^{2}+n x+m p=0, \\
\frac{1}{a} x^{2}+n x+... | 2. For any real numbers $m$, $n$, $p$, it is required:
(1) to write the corresponding quadratic equation;
(2) to make the equation have $n^{2}-4 m p$ as the discriminant.
Among the following answers:
$$
\begin{array}{l}
m x^{2}+n x+p=0, \\
p x^{2}+n x+m=0, \\
x^{2}+n x+m p=0, \\
\frac{1}{a} x^{2}+n x+a m p=0(a \neq 0)... | <think>
Okay, let's try to tackle this problem step by step. So, the question is about quadratic equations and their discriminants. The task has two parts: first, to write the corresponding quadratic equation given real numbers m, n, p, and second, to ensure that the discriminant of this equation is n² - 4mp. Then, amo... |
openr1_math_3830 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
$\textbf{(A)}\ 76\qquad \te... | Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
$\textbf{(A)}\ 76\qquad \textbf{(B)}\ 120\qq... | <think>
Okay, so I need to figure out the difference between the largest and smallest possible areas of rectangles that have integer side lengths and a perimeter of 50 units. Hmm, let me start by recalling some basic formulas related to rectangles.
First, the perimeter of a rectangle is calculated as P = 2*(length + w... |
openr1_math_3831 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2. If $3 \leqslant x+y \leqslant 5, -1 \leqslant x-y \leqslant 1$, then the maximum value of $3 x+y$ is<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the maximum value of 3x + y given the constraints 3 ≤ x + y ≤ 5 and -1 ≤ x - y ≤ 1. Hmm, let me think about how to approach this. Maybe... | 2. If $3 \leqslant x+y \leqslant 5, -1 \leqslant x-y \leqslant 1$, then the maximum value of $3 x+y$ is | <think>
Okay, so I need to find the maximum value of 3x + y given the constraints 3 ≤ x + y ≤ 5 and -1 ≤ x - y ≤ 1. Hmm, let me think about how to approach this. Maybe I can use inequalities or some sort of graphical method? Hmm.
First, let me write down the given inequalities:
1. 3 ≤ x + y ≤ 5
2. -1 ≤ x - y ≤ 1
And... |
openr1_math_3832 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
17. Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.)<|im_end|>
<|im_start|>assistant
<think>
Alright, so I need to find the largest natu... | 17. Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.) | <think>
Alright, so I need to find the largest natural number that can't be expressed as the sum of two composite numbers. Hmm, okay. Let me start by recalling what composite numbers are. They're natural numbers greater than 1 that aren't prime, right? So numbers like 4, 6, 8, 9, 10, etc. Prime numbers are 2, 3, 5, 7, ... |
openr1_math_3833 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
$29 \cdot 26\left(2^{48}-1\right)$ can be divided by two numbers between 60 and 70, these two numbers are
(A) 61,63 .
(B) 61,65 .
(C) 63,65 .
(D) 63,67 .
(E) 67,69 .
(22nd American High School Mathematics Examination, 1971)<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to solve this problem. ... | $29 \cdot 26\left(2^{48}-1\right)$ can be divided by two numbers between 60 and 70, these two numbers are
(A) 61,63 .
(B) 61,65 .
(C) 63,65 .
(D) 63,67 .
(E) 67,69 .
(22nd American High School Mathematics Examination, 1971) | <think>
Okay, let's try to solve this problem. So, we have the expression 29 times 26 times (2^48 minus 1), and we need to find which two numbers between 60 and 70 divide this expression. The options are given as pairs, so we have to check which pair of numbers are both divisors of this expression.
First, let's break ... |
openr1_math_3834 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x \cdot \cos x+\sin x}{(x \cdot \sin x)^{2}} d x
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the indefinite integral of (x·cosx + sinx) divided by (x·sinx)^2. Hmm, that looks a bit complicated, but maybe ther... | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x \cdot \cos x+\sin x}{(x \cdot \sin x)^{2}} d x
$$ | <think>
Okay, so I need to find the indefinite integral of (x·cosx + sinx) divided by (x·sinx)^2. Hmm, that looks a bit complicated, but maybe there's a substitution that can simplify it. Let me start by writing down the integral again to make sure I have it right:
∫ [x·cosx + sinx] / (x·sinx)^2 dx
First, let me simp... |
openr1_math_3835 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
4. Given the equation about $x$: $k x^{2}+\frac{|x|}{x+6}=0$ has four distinct real solutions. Then the range of the real number $k$ is $\qquad$ .<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to find the range of real numbers k such that the equation \( k x^{2} + \frac{|x|}{x + 6} = ... | 4. Given the equation about $x$: $k x^{2}+\frac{|x|}{x+6}=0$ has four distinct real solutions. Then the range of the real number $k$ is $\qquad$ . | <think>
Okay, let's see. I need to find the range of real numbers k such that the equation \( k x^{2} + \frac{|x|}{x + 6} = 0 \) has four distinct real solutions. Hmm. Alright, let's start by understanding what the equation looks like and how the solutions might come about.
First, let's rewrite the equation for clarit... |
openr1_math_3836 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
A triangle has sides $a, b, c$, and its area is $\frac{(a+b+c)(a+b-c)}{4}$. What is the measure of the largest angle of the triangle?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here: A triangle has sides a, b, c, and its area is ((a + b + c)(a + b - c))/4. I need to find the m... | A triangle has sides $a, b, c$, and its area is $\frac{(a+b+c)(a+b-c)}{4}$. What is the measure of the largest angle of the triangle? | <think>
Okay, so I have this problem here: A triangle has sides a, b, c, and its area is ((a + b + c)(a + b - c))/4. I need to find the measure of the largest angle of the triangle. Hmm, let's see. I should start by recalling some formulas related to the area of a triangle. The most common one is the Heron's formula, w... |
openr1_math_3837 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y \in \mathbb{R}$, we have:
$$
f(x+y)=f(x) f(y)
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find all continuous functions f from the real numbers to the real numbers such that for ... | Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y \in \mathbb{R}$, we have:
$$
f(x+y)=f(x) f(y)
$$ | <think>
Okay, so I need to find all continuous functions f from the real numbers to the real numbers such that for any real numbers x and y, f(x + y) equals f(x) times f(y). Hmm, functional equations, right? I remember hearing about these, and Cauchy's functional equation comes to mind. But this one is multiplicative r... |
openr1_math_3838 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
8. If the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-\mathrm{i}$, then $z_{1} \cdot z_{2}=$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to find the product of two complex numbers z₁ and z₂. The given information is t... | 8. If the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-\mathrm{i}$, then $z_{1} \cdot z_{2}=$ | <think>
Okay, let's see. I need to find the product of two complex numbers z₁ and z₂. The given information is that the magnitude of z₁ is 2, the magnitude of z₂ is 3, and they satisfy the equation 3z₁ - 2z₂ = 3/2 - i. Hmm, complex numbers with magnitude and an equation involving them. I remember that when dealing with... |
openr1_math_3839 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Task A-1.1.
Determine all pairs of natural numbers $(m, n)$ that satisfy the equation
$$
m(m-n)^{2}(m+n)=m^{4}+m n^{3}-99 n
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out how to solve this equation: m(m - n)²(m + n) = m⁴ + mn³ - 99n. Hmm, natural numbers m and n, so both... | ## Task A-1.1.
Determine all pairs of natural numbers $(m, n)$ that satisfy the equation
$$
m(m-n)^{2}(m+n)=m^{4}+m n^{3}-99 n
$$ | <think>
Okay, let me try to figure out how to solve this equation: m(m - n)²(m + n) = m⁴ + mn³ - 99n. Hmm, natural numbers m and n, so both are positive integers. Alright, let's start by expanding the left-hand side (LHS) and see if we can simplify the equation.
First, expand (m - n)². That's m² - 2mn + n². Then multi... |
openr1_math_3840 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 1. Solve the system of equations
$$
\frac{d y}{d x}=z, \frac{d z}{d x}=-y
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to solve this system of differential equations: dy/dx = z and dz/dx = -y. Hmm, so it's a system of two first-order equations. I remember that sometimes y... | Example 1. Solve the system of equations
$$
\frac{d y}{d x}=z, \frac{d z}{d x}=-y
$$ | <think>
Okay, let's see. I need to solve this system of differential equations: dy/dx = z and dz/dx = -y. Hmm, so it's a system of two first-order equations. I remember that sometimes you can convert a system like this into a single second-order differential equation. Let me try that approach.
Starting with the first ... |
openr1_math_3841 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
8,9
In a sphere of radius 9, three equal chords $A A 1, B B 1$, and $C C 1$ are drawn through a point $S$ such that $A S=4, A 1 S=8, B S < B 1 S, C S < C 1 S$. Find the radius of the sphere circumscribed about the pyramid $S A B C$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the r... | 8,9
In a sphere of radius 9, three equal chords $A A 1, B B 1$, and $C C 1$ are drawn through a point $S$ such that $A S=4, A 1 S=8, B S < B 1 S, C S < C 1 S$. Find the radius of the sphere circumscribed about the pyramid $S A B C$. | <think>
Okay, so I need to find the radius of the sphere circumscribed around the pyramid SABC, given that all these points are on a sphere of radius 9. The chords AA1, BB1, and CC1 pass through point S, and each chord is divided by S into segments of lengths AS=4, A1S=8 for the first chord. For the other two chords, B... |
openr1_math_3842 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last Aug... | [b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last August, Jennifer McL... | <think>
Okay, let's tackle problem p1. So, we have a right triangle with legs a and b, hypotenuse c. The perimeter is equal to the area. We need to prove there's only one possible value for a + b - c and find that value.
First, let's recall some basics. For a right triangle, by the Pythagorean theorem, a² + b² = c². T... |
openr1_math_3843 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2. The express passenger train, two days ago in the morning, departed from Gevgelija to Skopje, with several carriages in which there was a certain number of passengers. At the station in Veles, one passenger disembarked from the first carriage, two passengers disembarked from the last carriage, and no... | 2. The express passenger train, two days ago in the morning, departed from Gevgelija to Skopje, with several carriages in which there was a certain number of passengers. At the station in Veles, one passenger disembarked from the first carriage, two passengers disembarked from the last carriage, and no passengers board... | <think>
Okay, let's try to figure out this problem step by step. Hmm, it's about a train with passengers and some getting off at a station, then later another trip with fewer carriages and passengers. The question is asking for the number of passengers that started the journey in the morning from Gevgelija. Alright, le... |
openr1_math_3844 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
8.403. $\left\{\begin{array}{l}\sin x \sin y=0.75, \\ \tan x \tan y=3 .\end{array}\right.$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to solve this system of equations:
\[
\begin{cases}
\sin x \sin y = 0.75, \\
\tan x \tan y = 3.
\end{cases}
\]
Hmm, let's see. I remember that trigonometr... | 8.403. $\left\{\begin{array}{l}\sin x \sin y=0.75, \\ \tan x \tan y=3 .\end{array}\right.$ | <think>
Okay, so I need to solve this system of equations:
\[
\begin{cases}
\sin x \sin y = 0.75, \\
\tan x \tan y = 3.
\end{cases}
\]
Hmm, let's see. I remember that trigonometric equations often involve using identities to simplify them or express one function in terms of another. Let me start by looking at the two... |
openr1_math_3845 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. (3 points) In a certain country, there are 200 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layove... | 3. (3 points) In a certain country, there are 200 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layovers). What is the ... | <think>
Okay, let's see. The problem is about connecting 200 cities with flights operated by multiple airlines. The requirements are that every pair of cities is connected by exactly one airline's flight, and each airline's network must allow travel between any two cities (possibly with layovers). We need to find the m... |
openr1_math_3846 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
\section*{Problem 4 - 021244}
Given is a rectangle with sides \(2 a\) and \(2 b\), where \(a > b\). From this rectangle, four congruent right triangles (one at each corner, with the legs lying on the sides of the rectangle) are to be cut off such that the remaining figure forms an octagon with equal s... | \section*{Problem 4 - 021244}
Given is a rectangle with sides \(2 a\) and \(2 b\), where \(a > b\). From this rectangle, four congruent right triangles (one at each corner, with the legs lying on the sides of the rectangle) are to be cut off such that the remaining figure forms an octagon with equal sides.
The side o... | <think>
Okay, let me try to figure out this problem. So, we have a rectangle with sides 2a and 2b, where a is longer than b. We need to cut off four congruent right triangles from each corner, such that the remaining figure is an octagon with all sides equal. The goal is to express the side length of the octagon in ter... |
openr1_math_3847 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
[
doubling the median ] [Theorem of the sum of the squares of the diagonals ]
In a triangle, two sides are equal to 11 and 23, and the median drawn to the third side is 10. Find the third side.
#<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the third side of a triangle where two s... | [
doubling the median ] [Theorem of the sum of the squares of the diagonals ]
In a triangle, two sides are equal to 11 and 23, and the median drawn to the third side is 10. Find the third side.
# | <think>
Okay, so I need to find the third side of a triangle where two sides are 11 and 23, and the median to the third side is 10. Hmm. Let me recall what I know about medians in triangles. The median is the line segment from a vertex to the midpoint of the opposite side. There's a formula related to the lengths of th... |
openr1_math_3848 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
6. (8 points) On the board, 33 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 33 minutes?<|im_end|>
<|i... | 6. (8 points) On the board, 33 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 33 minutes? | <think>
Okay, let's see. So the problem is about Karlson erasing two numbers and writing their sum back on the board, and each time he does that, he eats candies equal to the product of the two numbers he erased. We start with 33 ones, and he does this for 33 minutes. We need to find the maximum number of candies he co... |
openr1_math_3849 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Find the number of different sequences with $2 n+1$ terms, where each term is $\pm 1$ and the absolute value of the sum of any odd number of consecutive terms does not exceed 1.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation re... | Find the number of different sequences with $2 n+1$ terms, where each term is $\pm 1$ and the absolute value of the sum of any odd number of consecutive terms does not exceed 1.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | <think>
Okay, let's see. The problem is to find the number of different sequences with 2n+1 terms, where each term is either +1 or -1. The condition is that the absolute value of the sum of any odd number of consecutive terms doesn't exceed 1. So, for any odd-length consecutive subsequence, the sum must be either -1, 0... |
openr1_math_3850 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Task 3 - 030723
A wooden cube with an edge length of $30 \mathrm{~cm}$ is to be cut into cubes with an edge length of $10 \mathrm{~cm}$.
a) How many cuts must be made? (Sawing in a package is not allowed.)
b) How many cubes will you get?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to ... | ## Task 3 - 030723
A wooden cube with an edge length of $30 \mathrm{~cm}$ is to be cut into cubes with an edge length of $10 \mathrm{~cm}$.
a) How many cuts must be made? (Sawing in a package is not allowed.)
b) How many cubes will you get? | <think>
Okay, so I need to solve part a) and b) of this problem about cutting a wooden cube. Let me start by understanding the problem thoroughly.
The original cube has an edge length of 30 cm. We need to cut it into smaller cubes with edge length 10 cm. The question is asking how many cuts are required and how many s... |
openr1_math_3851 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
5. In $\triangle A B C$, $\angle B A C=60^{\circ}, \angle A O B=\angle B O C=\angle A O C$. If $A O+B O+C O=57$, $A O: B O=2: 3$, then $C O=$ $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out this problem. So, we have triangle ABC with angle BAC of 60 degrees. The angles A... | 5. In $\triangle A B C$, $\angle B A C=60^{\circ}, \angle A O B=\angle B O C=\angle A O C$. If $A O+B O+C O=57$, $A O: B O=2: 3$, then $C O=$ $\qquad$ | <think>
Okay, let me try to figure out this problem. So, we have triangle ABC with angle BAC of 60 degrees. The angles AOB, BOC, and AOC are all equal. Hmm, that means each of those angles is 120 degrees because there are three of them and they must add up to 360 degrees around point O. Wait, but in a triangle, the sum... |
openr1_math_3852 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Find all triplets of strictly positive integers $(a, b, c)$ such that $6^{a}=1+2^{b}+3^{c}$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find all triplets of strictly positive integers (a, b, c) such that 6^a = 1 + 2^b + 3^c. Hmm, let's start by understanding what this equation is saying.... | Find all triplets of strictly positive integers $(a, b, c)$ such that $6^{a}=1+2^{b}+3^{c}$ | <think>
Okay, so I need to find all triplets of strictly positive integers (a, b, c) such that 6^a = 1 + 2^b + 3^c. Hmm, let's start by understanding what this equation is saying. The left side is 6 raised to some power, which is a combination of 2s and 3s, and the right side is 1 plus a power of 2 and a power of 3. Si... |
openr1_math_3853 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
13. (10 points) In the equation below, $A, B, C, D, E, F, G, H, I$ each represent different digits from $1 \sim 9$.
$$
\overline{\mathrm{ABCD}}+\overline{\mathrm{EF}} \times \overline{\mathrm{GH}}-I=X
$$
Then the minimum value of $X$ is . $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's ta... | 13. (10 points) In the equation below, $A, B, C, D, E, F, G, H, I$ each represent different digits from $1 \sim 9$.
$$
\overline{\mathrm{ABCD}}+\overline{\mathrm{EF}} \times \overline{\mathrm{GH}}-I=X
$$
Then the minimum value of $X$ is . $\qquad$ | <think>
Okay, let's tackle this problem step by step. So, we need to find the minimum value of X in the equation:
ABCD + EF × GH - I = X
where each letter A, B, C, D, E, F, G, H, I represents a different digit from 1 to 9. Our goal is to assign these digits such that X is as small as possible. Since all digits must b... |
openr1_math_3854 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
5. Inside an isosceles triangle $A B C (A B=A C)$, a point $K$ is marked. Point $L$ is the midpoint of segment $B K$. It turns out that $\angle A K B=\angle A L C=90^{\circ}, A K=C L$. Find the angles of triangle $A B C$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to solve this geometry p... | 5. Inside an isosceles triangle $A B C (A B=A C)$, a point $K$ is marked. Point $L$ is the midpoint of segment $B K$. It turns out that $\angle A K B=\angle A L C=90^{\circ}, A K=C L$. Find the angles of triangle $A B C$. | <think>
Okay, let me try to solve this geometry problem. Hmm, so we have an isosceles triangle ABC with AB = AC. There's a point K inside the triangle, and L is the midpoint of BK. We know that angle AKB and angle ALC are both 90 degrees, and AK equals CL. We need to find the angles of triangle ABC. Alright, let's star... |
openr1_math_3855 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
5. (8 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $12$, and $\cos \alpha = \frac{3}{4}$?<|im_en... | 5. (8 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $12$, and $\cos \alpha = \frac{3}{4}$? | <think>
Okay, let's see. I need to find the length of BC in this circle problem. The circle has center O and radius 12. Point M is on radius AO. On one side of AO, points B and C are chosen such that angles AMB and OMC are both equal to alpha, and we know that cos(alpha) is 3/4. Hmm. Let me visualize this.
First, let... |
openr1_math_3856 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2.1. Determine the number of natural divisors of the number $11!=1 \cdot 2 \cdot \ldots \cdot 10 \cdot 11$ that are multiples of three. Answer. 432.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to figure out how many natural divisors of 11! are multiples of three. Hmm, alright. Let's start b... | 2.1. Determine the number of natural divisors of the number $11!=1 \cdot 2 \cdot \ldots \cdot 10 \cdot 11$ that are multiples of three. Answer. 432. | <think>
Okay, so I need to figure out how many natural divisors of 11! are multiples of three. Hmm, alright. Let's start by recalling that 11! means 11 factorial, which is the product of all natural numbers from 1 to 11. So, 11! = 1×2×3×...×10×11.
First, I think the key here is to find the prime factorization of 11!.... |
openr1_math_3857 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
2.033. $\frac{\sqrt{(2 p+1)^{3}}+\sqrt{(2 p-1)^{3}}}{\sqrt{4 p+2 \sqrt{4 p^{2}-1}}}$.
2.033. $\frac{\sqrt{(2 p+1)^{3}}+\sqrt{(2 p-1)^{3}}}{\sqrt{4 p+2 \sqrt{4 p^{2}-1}}}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this expression to simplify: the numerator is the sum of the square roots ... | 2.033. $\frac{\sqrt{(2 p+1)^{3}}+\sqrt{(2 p-1)^{3}}}{\sqrt{4 p+2 \sqrt{4 p^{2}-1}}}$.
2.033. $\frac{\sqrt{(2 p+1)^{3}}+\sqrt{(2 p-1)^{3}}}{\sqrt{4 p+2 \sqrt{4 p^{2}-1}}}$. | <think>
Okay, so I have this expression to simplify: the numerator is the sum of the square roots of (2p + 1) cubed and (2p - 1) cubed, and the denominator is the square root of (4p + 2 times the square root of (4p squared minus 1)). Hmm. Let me write that down again to make sure I have it right.
Numerator: √[(2p + 1)... |
openr1_math_3858 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 1 As shown in Figure 1, in quadrilateral $A B C D$, $A C$ and $B D$ are diagonals, $\triangle A B C$ is an equilateral triangle, $\angle A D C=30^{\circ}, A D=3, B D=$ 5. Then the length of $C D$ is ( ). ${ }^{[1]}$
(A) $3 \sqrt{2}$
(B) 4
(C) $2 \sqrt{5}$
(D) 4.5<|im_end|>
<|im_start|>assistant... | Example 1 As shown in Figure 1, in quadrilateral $A B C D$, $A C$ and $B D$ are diagonals, $\triangle A B C$ is an equilateral triangle, $\angle A D C=30^{\circ}, A D=3, B D=$ 5. Then the length of $C D$ is ( ). ${ }^{[1]}$
(A) $3 \sqrt{2}$
(B) 4
(C) $2 \sqrt{5}$
(D) 4.5 | <think>
Okay, so I need to find the length of CD in this quadrilateral ABCD. Let me see. The problem says that triangle ABC is equilateral, angle ADC is 30 degrees, AD is 3, BD is 5. Hmm. Let me try to visualize this.
First, since ABC is an equilateral triangle, all sides are equal. So AB = BC = AC. But the problem d... |
openr1_math_3859 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. The Mad Hatter's clock is fast by 15 minutes per hour, while the March Hare's clock is slow by 10 minutes per hour. One day, they set their clocks by the Dormouse's clock (which is stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for their traditional five o'clock tea. ... | 1. The Mad Hatter's clock is fast by 15 minutes per hour, while the March Hare's clock is slow by 10 minutes per hour. One day, they set their clocks by the Dormouse's clock (which is stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for their traditional five o'clock tea. How long will the... | <think>
Okay, let's see. The problem is about the Mad Hatter and the March Hare having clocks that run fast and slow respectively. They set their clocks to the Dormouse's clock, which is stopped at 12:00. Then they agree to meet at 5 o'clock, which is 17:00. The question is, how long will the Mad Hatter wait for the Ma... |
openr1_math_3860 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Task 3 - 050723
Compare the sum of all three-digit natural numbers divisible by 4 with the sum of all three-digit even natural numbers not divisible by 4!
a) Which of the two sums is greater?
b) What is the difference between the two sums in absolute value?<|im_end|>
<|im_start|>assistant
<think>... | ## Task 3 - 050723
Compare the sum of all three-digit natural numbers divisible by 4 with the sum of all three-digit even natural numbers not divisible by 4!
a) Which of the two sums is greater?
b) What is the difference between the two sums in absolute value? | <think>
Okay, so I need to compare two sums: one is the sum of all three-digit natural numbers divisible by 4, and the other is the sum of all three-digit even natural numbers not divisible by 4. Then I have to determine which sum is greater and find the absolute difference between them. Let me break this down step by ... |
openr1_math_3861 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. Find such an integer $\mathrm{n}$ that for any integer $\mathrm{k} \neq 2013$, the number $n-k^{2014}$ is divisible by $2013-k$.
Answer: $2013^{2014}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here: I need to find an integer n such that for any integer k not equal to 201... | 3. Find such an integer $\mathrm{n}$ that for any integer $\mathrm{k} \neq 2013$, the number $n-k^{2014}$ is divisible by $2013-k$.
Answer: $2013^{2014}$. | <think>
Okay, so I have this problem here: I need to find an integer n such that for any integer k not equal to 2013, the number n - k^2014 is divisible by 2013 - k. And the answer is supposed to be 2013^2014, but I need to figure out why. Let me try to work through this step by step.
First, let's parse the problem. W... |
openr1_math_3862 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer.
[i]Proposed by Oleksiy Masalitin[/i]<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find all pairs of positive integers (m, n) such that 2^n - 13^m is a cube of a positive integer. ... | Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer.
[i]Proposed by Oleksiy Masalitin[/i] | <think>
Okay, so I need to find all pairs of positive integers (m, n) such that 2^n - 13^m is a cube of a positive integer. Let me start by understanding the problem better.
We have the equation 2^n - 13^m = k^3, where k is a positive integer. So, both m and n have to be positive integers, and the result of subtractin... |
openr1_math_3863 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
1. The range of the function $f(x)=\sin ^{4} x \tan x+\cos ^{4} x \cot x$ is<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the range of the function \( f(x) = \sin^4 x \tan x + \cos^4 x \cot x \). Hmm, let's start by understanding what this function looks like. Maybe I can simplify it... | 1. The range of the function $f(x)=\sin ^{4} x \tan x+\cos ^{4} x \cot x$ is | <think>
Okay, so I need to find the range of the function \( f(x) = \sin^4 x \tan x + \cos^4 x \cot x \). Hmm, let's start by understanding what this function looks like. Maybe I can simplify it first. Let's see, both terms have trigonometric functions raised to the fourth power multiplied by tangent or cotangent. Let ... |
openr1_math_3864 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
7. The function
$$
f(x)=x+\cos x-\sqrt{3} \sin x
$$
passes through a point on its graph where the slope of the tangent line is $k$. Then the range of values for $k$ is $\qquad$ .<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here where I need to find the range of possible values... | 7. The function
$$
f(x)=x+\cos x-\sqrt{3} \sin x
$$
passes through a point on its graph where the slope of the tangent line is $k$. Then the range of values for $k$ is $\qquad$ . | <think>
Okay, so I have this problem here where I need to find the range of possible values for k, which is the slope of the tangent line to the function f(x) = x + cos x - sqrt(3) sin x at some point on its graph. Hmm, let me think about how to approach this.
First, I remember that the slope of the tangent line to a ... |
openr1_math_3865 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
10. According to the pattern of the following equations, find the sum of the 2018th equation.
$$
2+3,3+7,4+11,5+15,6+19 \ldots
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is asking me to find the sum of the 2018th equation in the given pattern. The equations listed are: 2+3,... | 10. According to the pattern of the following equations, find the sum of the 2018th equation.
$$
2+3,3+7,4+11,5+15,6+19 \ldots
$$ | <think>
Okay, let's see. The problem is asking me to find the sum of the 2018th equation in the given pattern. The equations listed are: 2+3, 3+7, 4+11, 5+15, 6+19, and so on. So first, I need to figure out the pattern here and then determine what the 2018th equation would be and its sum.
Looking at the first few equa... |
openr1_math_3866 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$,
and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is
$\textbf{(A) }6\qquad \textbf{(B) }2\sqrt{6}\qquad \textbf{(C) }\frac{8\s... | If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$,
and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is
$\textbf{(A) }6\qquad \textbf{(B) }2\sqrt{6}\qquad \textbf{(C) }\frac{8\sqrt{3}}{3}\qquad ... | <think>
Okay, let's try to figure out this problem step by step. It's about a regular hexagon with apothem 2, and we need to find the area of a quadrilateral formed by the midpoints of four of its sides. The answer choices are given, so that might help check our work along the way.
First, let me recall some properties... |
openr1_math_3867 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
## Task B-2.7.
Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by,
$$
f(x)=\frac{2}{\sqrt[3]{x^{2}+4 x+4}+\sqrt[3]{x^{2}-4}+\sqrt[3]{x^{2}-4 x+4}}
$$
What is $f(4)+f(8)+f(12)+\cdots+f(2024)$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is asking for the sum ... | ## Task B-2.7.
Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by,
$$
f(x)=\frac{2}{\sqrt[3]{x^{2}+4 x+4}+\sqrt[3]{x^{2}-4}+\sqrt[3]{x^{2}-4 x+4}}
$$
What is $f(4)+f(8)+f(12)+\cdots+f(2024)$? | <think>
Okay, let's see. The problem is asking for the sum of f(4) + f(8) + f(12) + ... + f(2024), where the function f(x) is defined as 2 divided by the sum of three cube roots. The cube roots are of x² + 4x + 4, x² - 4, and x² - 4x + 4. Hmm, that looks a bit complicated, but maybe there's a pattern or simplification ... |
openr1_math_3868 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
5.35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$.
(2) Prove that this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem to solve. It's ab... | 5.35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$.
(2) Prove that this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$. | <think>
Okay, so I have this problem to solve. It's about finding the minimum value of the polynomial \( P(x, y) = 4 + x^2 y^4 + x^4 y^2 - 3 x^2 y^2 \), and then proving that this polynomial can't be written as a sum of squares of other polynomials. Let's start with part (1).
First, I need to find the minimum value of... |
openr1_math_3869 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass bet... | A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at ... | <think>
Okay, let me try to figure out this problem. So, there's a car and a minivan going from Alphaville to Betatown. The car's speed is 40 km/h, and the minivan's speed is 50 km/h. The minivan passes the car 10 minutes before the car arrives at Betatown. We need to find how many minutes pass between the minivan arri... |
openr1_math_3870 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Problem 6. Find the number of non-empty sets of $S_{n}=\{1,2, \ldots, n\}$ such that there are no two consecutive numbers in one and the same set.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the number of non-empty subsets of the set S_n = {1, 2, ..., n} such that no two consecutive... | Problem 6. Find the number of non-empty sets of $S_{n}=\{1,2, \ldots, n\}$ such that there are no two consecutive numbers in one and the same set. | <think>
Okay, so I need to find the number of non-empty subsets of the set S_n = {1, 2, ..., n} such that no two consecutive numbers are in the same subset. Hmm, let me think about how to approach this.
First, let me make sure I understand the problem correctly. We need all subsets where if you pick any two numbers fr... |
openr1_math_3871 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
11. Given the polar equation of a line is $\rho \sin \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$, then the distance from the pole to the line is $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this polar equation of a line: ρ sin(θ + π/4) = √2/2. And the question is asking fo... | 11. Given the polar equation of a line is $\rho \sin \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$, then the distance from the pole to the line is $\qquad$ | <think>
Okay, so I have this polar equation of a line: ρ sin(θ + π/4) = √2/2. And the question is asking for the distance from the pole to the line. Hmm, let me think. I remember that in polar coordinates, the general form of a line is ρ sin(θ - α) = d, where d is the distance from the pole to the line, and α is the an... |
openr1_math_3872 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the maximum value of the expression S = (xy + 2yz + zw) / (x² + y² + z² + w²), where x, y, z, w are rea... | Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value. | <think>
Okay, so I need to find the maximum value of the expression S = (xy + 2yz + zw) / (x² + y² + z² + w²), where x, y, z, w are real numbers and not all zero. Hmm, this looks like a problem where I have to maximize a quadratic form over the Euclidean norm. Maybe I can use some linear algebra techniques here.
Firs... |
openr1_math_3873 | DeepSeek-R1-OpenR1-Math-220k | mathematics | <|im_start|>user
3. Solve the system of equations:
$$
\left\{\begin{aligned}
x+y-2018 & =(y-2019) \cdot x \\
y+z-2017 & =(y-2019) \cdot z \\
x+z+5 & =x \cdot z
\end{aligned}\right.
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I have this system of equations to solve. Let me write them down again to make... | 3. Solve the system of equations:
$$
\left\{\begin{aligned}
x+y-2018 & =(y-2019) \cdot x \\
y+z-2017 & =(y-2019) \cdot z \\
x+z+5 & =x \cdot z
\end{aligned}\right.
$$ | <think>
Okay, let's see. I have this system of equations to solve. Let me write them down again to make sure I have them right:
1. \( x + y - 2018 = (y - 2019) \cdot x \)
2. \( y + z - 2017 = (y - 2019) \cdot z \)
3. \( x + z + 5 = x \cdot z \)
Hmm. So three equations with three variables: x, y, z. I need to find val... |
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