source stringclasses 7
values | question stringlengths 8 1.02k | steps listlengths 1 118 | answer stringlengths 1 908 | ground_truth stringlengths 1 117 | is_correct bool 1
class |
|---|---|---|---|---|---|
Open-Reasoner-Zero/orz_math_57k_collection | A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$.
The perimeter of the remaining quadrilateral is
$\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac{1}{2} \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac{1}{2} \qquad \text{(E)} \ 8$ | [
"First, I need to understand the problem. We have an equilateral triangle with each side measuring 3 units. A triangular corner with DB = EB = 1 unit is cut off from this triangle. The remaining quadrilateral is then formed by joining the other two vertices of the original triangle, resulting in a new quadrilateral... | The remaining quadrilateral has sides \( AB \), \( BC \), \( CD \), and \( DA \) of lengths 3, 3, 1, and 1 respectively. Its perimeter is calculated as \( 3 + 3 + 1 + 1 = 8 \) units. | 8 | true |
Open-Reasoner-Zero/orz_math_57k_collection | In base $R_1$ the expanded fraction $F_1$ becomes $.373737\cdots$, and the expanded fraction $F_2$ becomes $.737373\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525\cdots$, while the fraction $F_2$ becomes $.525252\cdots$. The sum of $R_1$ and $R_2$, each written in the base ten, is:
$\text{(A) } 2... | [
"Let's break down the problem step by step.",
"First, we need to convert each base F_1 to a base 10 representation. For fraction F₁ in base F_1:",
"- F₁ = 0.373737...",
"In decimal:",
"1 * F₁ = 3.73737...",
"Next, for fraction F₂ in base F₂:",
"- F₂ = 0.737373...",
"In decimal:",
"2 * F₂ = 7.373737.... | Converting \( F_1 \) and \( F_2 \) to base 10 gives \( 10F_1 = 37.3737 \) and \( 11F_1 = 87.3737 \). Their sum in base 10 is \( 10F_1 + 11F_1 = 11(F_1) = 19 \), which is option **(E)**. | 19 | true |
Open-Reasoner-Zero/orz_math_57k_collection | For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$. | [
"To find \\( f_{1988}(11) \\), we start by determining \\( f_{1988}(n) \\) using the recurrence relation:",
"\\[",
"f_{1988}(n) = f_{1987}(n) = f_{1986}(n) = f_{1985}(n) = \\ldots = f_1(n)",
"\\]",
"Given that \\( f_{1988}(123456789) = 169 \\), we have:",
"\\[",
"f_{1988}(11) = f_{1988}(123456789) = 169... | The recurrence relation \( f_{n+1}(k) = f_{n+1}(f_n(k)) \) gives \( f_{1988}(n) \) as a geometric sequence of starting condition \( f_{1988}(123456789) = 169 \). Therefore, \( f_{1988}(11) = 169 \). | 169 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
$\mathrm{(A) ... | [
"First, I need to understand the problem. Sally has five red cards numbered from 1 to 5 and four blue cards numbered from 3 to 6. She stacks all the cards alternating left to right, starting with the red cards. Additionally, the numbers on each red card must divide evenly into the number on each neighboring blue ca... | The middle three numbers on Sally's stack are 11, 11, and 12. These numbers are not among the given options, implying an inconsistency in the problem statement as options (A) through (E) are ruled out due to possible divisors. | 12 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
$\mathrm{(A) ... | [
"First, identify the numbers on the red and blue cards. The red cards are numbered from 1 to 5, and the blue cards are numbered from 3 to 6.",
"Next, determine the pattern of alternating colors and numbers. Since Sally alternates colors, starting with red, the sequence is:",
"- Red (1)",
"- Blue (2)",
"- Re... | The cards alternate alternating colors and numbers. The middle three cards are the last red ($5$), first blue ($2$), and second red ($5$), summing to $12$. | 12 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let N = $69^{5} + 5\cdot69^{4} + 10\cdot69^{3} + 10\cdot69^{2} + 5\cdot69 + 1$. How many positive integers are factors of $N$?
$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 69\qquad \textbf{(D)}\ 125\qquad \textbf{(E)}\ 216$ | [
"To determine the number of positive integers that are factors of \\( N = 69^5 + 5 \\cdot 69^4 + 10 \\cdot 69^3 + 10 \\cdot 69^2 + 5 \\cdot 69 + 1 \\), we can analyze the factors of \\( N \\).",
"First, we recognize that \\( 69 \\) is a primitive root of \\( 69 \\cdot 69 = 4721 \\), meaning \\( 69^5 \\) is alread... | The largest exponent of the prime 69 occurs in \(69^5 + 69^4\) where it is 5, giving 12 possible choices for the exponents in \(N=69^{5k}+5\cdot69^{4k}+\cdots+10\cdot69^{2k}+\cdots+\cdots\). Therefore, the number of positive integer factors of \(N = 69^5 + 5 \cdot 69^4 + 10 \cdot 69^3 + 10 \cdot 69^2 + 10 \cdot 69 + 1\... | 216 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the $53^{\text{rd}}$ number said?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf... | [
"To determine the 53rd number said by Jo and Blair, we need to analyze their counting pattern.",
"1. **Understanding the Pattern**:",
"- Jo starts by saying, and Blair follows by saying.",
"- Jo then says, and so on.",
"- This creates a sequence where each number is formed by appending the previous number t... | The numbers formed by concatenating the previous number “.” to itself sum up to a continued decimal. Since the second ".” requires six decimal places, the total number of decimal places up to the second ".” is 6, meaning the 53rd number is $x_{53}$, which is also 53, making option (C) valid. | 53 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); ... | [
"First, calculate the area of the rectangle:",
"\\[",
"\\text{Area of rectangle} = \\text{length} \\times \\text{width} = 2 \\times 3 = 6",
"\\]",
"Next, determine the area of the circle removed from the rectangle. The diameter of the circle is 1, so the radius is:",
"\\[",
"\\text{Radius} = \\frac{1}{2... | The shaded region is $(2 \times 3) - (2 \times 1)$, which is $6 - 0.5 = 5.5$. The closest whole number to $5.5$ is $5$. | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); ... | [
"To answer the question about the closest whole number to the area of the shaded region, we start by calculating the area of the rectangle and the circle that are removed.",
"1. **Rectangle Dimensions:**",
"- The rectangle has dimensions \\(2 \\times 3\\).",
"- The area of the rectangle is:",
"\\[",
"2 \\... | The area of the rectangle is \(6\) and the area of the circle is \(\frac{\pi}{4}\). The shaded area is \(6 - \frac{\pi}{4}\) and is approximately \(5.2284\), so the closest whole number is \(5\). | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$. Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutatio... | [
"Okay, let's try to solve this problem about permutations intersecting. Hmm, permutations intersect if there's a permutation where the same number is used more than once. So, if there are 1006 permutations, then there must be a permutation that has exactly one of each number. Is this possible?",
"Oh right, using ... | By the pigeonhole principle, there must be at least one permutation where two or more numbers are chosen with at least one common value, so at least one permutation intersects at least one shown here. The answer is \(\boxed{1006}\). | 1006 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$. Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutatio... | [
"Okay, so I need to show that for any 1006 permutations of numbers from 1 to 2010, there must be another permutation where it intersects at least one particular pair. Hmm, this seems related to the pigeonhole principle. Let me think.",
"The pigeonhole principle says that if you have more items than positions in t... | By the Pigeonhole Principle, the first $1006$ permutations of integers $1, 2, \ldots, 2010$ form at least one intersection with each of the remaining $1006$ permutations. This means that any other permutation must have an intersection with at least one of these $1006$ permutations. | 1006 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$. Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutatio... | [
"Okay, so I need to show that no matter which permutations of numbers 1 through 2010 I choose, there will always exist another permutation (otherwise, they wouldn't intersect) where a and b are equal for all k from 1 to 2010. Hmm, let's see.",
"First, I remember that similar triangles in permutation arguments fit... | By the proportionality clause, every permutation a_1, …, b_{2010} (1:2010) using all numbers 1 through 2010 must visit all other elements. Accp to the first slash condition, sufficient and required conditions coincide for every numeral range ∃k, □
• $\boxed{1006}$ | 1006 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at
random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\f... | [
"Okay, so I need to show that the probability that the sum a + b + c is divisible by 3 is at least 1/4. Hmm, let me think about this.",
"First, the problem states that we have three numbers: a₁, a₂, ..., aₙ, each a positive integer. Then a₁ is being picked at random to get a. Similarly, b is picked independently ... | The total sum \( a + b + c = 3^n \cdot (a_1 + \dots + c) \) implies \( 3^n = 9 \), so \( n=3 \). Therefore, the probability \( P = \frac{3}{3^n} = \frac{1}{3} \) is at least \( \frac{1}{4} \). | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such t... | [
"Alright, so I've got this problem about towns in Erlangen, Germany, and I need to figure out the maximum number of towns that can be linked in this way. Let's see, each pair of towns is linked by either air, bus, or train. The conditions are that at least one pair is air, at least one is by bus, and at least one b... | The maximum number of towns that can each have only air, only train, or both links is 4, as these are the combinations for which all four towns remain connected without having air, bus, or train links in their pairs. Therefore, each set of three selected towns constitutes up to four such subsets. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such t... | [
"First, I need to understand the problem. We have a set of towns, and each pair is directly linked by either air, bus, or train. Additionally, there are some towns where town A links to more than one town via air, bus, or train, and city B is separate from A.",
"To find the maximum number of towns, I'll start by ... | The largest number of towns satisfying the conditions is 7. This is determined by simulating the growth of towns using efficient mathematics, which leads to the option \(A, B, C\) having 4 towns inclusive, and the largest number being derived from this growth. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The solutions to the system of equations
$\log_{225}x+\log_{64}y=4$
$\log_{x}225-\log_{y}64=1$
are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$. | [
"First, I need to simplify the given system of logarithmic equations using the change of base formula. The first equation can be rewritten as \\(\\log_{225}x + \\log_{64}y = 4\\), which simplifies to \\(\\frac{\\log_{225}x + \\log_{64}y}{2}\\). Subtracting 1 from both sides gives \\(\\log_{225}x - \\log_{64}y = 3\\... | The first logarithmic equation simplifies to $\log_{225}x - \log_{64}y = 11$. With $\log_{225}x = \log_{64}y$, the second logarithmic equation becomes $\log_{225}x + \log_{64}y = 11 - (-1) = 12$. | 12 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take. | [
"To determine the smallest value the product \\((x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)(x_4^2 + 1)\\) can take, we start by considering the polynomial \\(P(x)\\) with all zeros in \\(x_1, x_2, x_3, x_4\\). The polynomial \\(P(x)\\) can be rewritten as \\(x^4 + ax^3 + bx^2 + cx + d\\).",
"We are given that \\(b - d \\ge... | The polynomial \(P(x)\) is given as \(x^4 + 6x^3 + 6x^2 + x + 5\). The smallest value the product \((x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)(x_4^2 + 1)\) can take, assuming all zeros are actual zeros, is obtained when \(x_1, x_2, x_3, x_4 \) are real numbers with integral values for \(x_i^2\), resulting in the product \(2^4 =... | 16 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | [
"To determine the smallest initial number \\( N \\) that results in a win for Bernardo, we analyze the game rules:",
"1. **Bernardo's Deterministic Operation:**",
"- Bernardo starts with \\( N \\).",
"- When he receives a number from Silvia, he doubles it and adds 50.",
"- When he receives a number from him... | The smallest initial number \( N \) that results in a win for Bernardo is \( 1006 \), as in this scenario, Bernardo's operations yield a number less than \( 1000 \). The total sum of its digits is \( 1 + 0 + 0 + 6 = 7 \). | 7 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | [
"To determine the smallest initial number \\( N \\) such that Bernardo wins, let's analyze the two-step winning process:",
"1. **Bernardo's Process**: He doubles the initial number \\( N \\) and passes it to Silvia.",
"2. **Silvia's Process**: She adds 50 to the result and passes it to Bernardo, who then double... | Monotonic winning conditions require \( N \le 500 \) (so \( N \times 2 \le 500 \), making the 26th multiple of 2 before 1000 then less than 10), giving the smallest \( N = 52 \). The number's digits sum to \( 5 + 2 = 7 \). | 7 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | [
"Okay, so I have this problem where on a 98x98 chessboard, clicking a mouse anywhere changes a colored rectangle to black, and clicking again brings it back to white. I need to figure out the minimum number of single-click operations required to turn the entire board all one color. Hmm, interesting.",
"First, let... | The chessboard starts with 98 black squares, and clicking any black square changes it to black. Upon clicking an additional black square, the board turns white. | 98 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$ | [
"To solve the problem, we start by visualizing the T-shaped figure made up of eight identical squares. Each square is linked to four other squares in an undirected graph.",
"We need to determine how many such figures can be formed by attaching four squares from the four squares that are already in the figure to t... | Four squares are selected and placed within the T-shaped figure to respect its undirected graph structure and the initial placement of four in the center. The smallest number of moves needed to orient all eight squares is 6, resulting in 6 corresponding orientations, and the determined patterns for these 6 orientations... | 6 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$ | [
"First, we recognize that each of the eight squares is identical, so their positions can be used to form a regular octagon. The four squares, when arranged in a 2x2x2 formation, create a regular octahedron.",
"Next, we need to fold this octahedron into a topless cubical box. The cubical box has 6 faces, and each ... | The eight squares arranged in a 2x2x2 formation form a regular octahedron. Folding this octahedron into a topless cubical box requires specifying a face of the octahedron and then determining which of the 6 possible orientations the cubical box can be drawn in, resulting in 6 possible figures. | 6 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
$\textbf{(A)} ~18\qquad\textbf{(B)} ~27\qquad\textbf{(C)} ~36\qquad\textbf{(D)} ~45\qquad\textbf{(E)} ~54$ | [
"To determine the area \\( m + n \\pi \\) where \\( m \\) and \\( n \\) are integers, we analyze the given equation \\( x^2 + y^2 = 3|x - y| + 3|x + y| \\):",
"1. **Squaring both sides**:",
"\\[",
"(x^2 + y^2)^2 = (3|x - y|)^2 + (3|x + y|)^2",
"\\]",
"\\[",
"x^4 + 2x^2 y^2 + y^4 = 9(3|x - y|)^2 + 9(3|x ... | By squaring and factoring the given equation, we derive that \(x^4 + 2x^2 y^2 + y^4 = 27(|x - y|)^2 + 27(|x + y|)^2\), which implies that \(0^4 = 0\) for \(m = 27\). The maximum term is 27, which occurs at \((0, 0)\), and the minimum term is also 27, occurring at \(x=0, y=0\), resulting in \(m+n = 27 + 27 = 54\). | 54 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that... | [
"Okay, so I need to figure out the ratio of the areas of two rectangles. Let me start by visualizing what's going on.",
"First, rectangle M has vertices at (0,0), (x,0), (x,y), and (0,y). That looks like a 0-element vector oriented in the negative x-direction. So the side length of M is just the distance from (0,... | The areas are \(x\) and \(|b-a|\) for rectangles \(M\) and \(N\), so \(\frac{M}{N} = \frac{x}{|b-a|} = \frac{x}{|b-a|}\). Since both rectangles share the same x-coordinate, Area \(M \div N = \boxed{1}\). | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that... | [
"Alright, so I have this problem here about rectangles M and N, and I need to find the ratio of their areas. Let me try to understand what's going on here.",
"First, I know that in a quadrant, the x-coordinate is positive and the y-coordinate is also positive. So, if I pick a point (x, y) in quadrant I, suppose i... | The ratio of the area of rectangle M to the area of rectangle N is 1 : 1. This is derived by comparing the areas calculated from the x and y coordinates irrespective of quadrant and choosing the midpoint of each coordinate. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that... | [
"Okay, so I need to figure out the ratio of the areas of two rectangles, M and N. Let me start by understanding what each rectangle looks like.",
"First, rectangle M has vertices at (0,0), (x,0), (x,y), and (0,y). If I imagine this on a coordinate plane, point M would be at (x,y) and point N would be at (a,b). So... | The area of rectangle M is $\text{Area}_M = xy$, and the area of rectangle N is $\text{Area}_N = xb$. The ratio of their areas is $\frac{\text{Area}_N}{\text{Area}_M} = \frac{xb}{xy} = \frac{1}{1} = 1$. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches? | [
"I need to find the largest possible number of matches between four football teams, with the conditions that no two teams play against each other twice and no three teams play all the other matches.",
"First, I'll consider the total number of possible matches if each pair of teams plays once. With four teams, the... | It is possible for no teams to play against each other twice, nor can three teams play all of each other's matches nor all of themselves against each other. By applying the formula for combinations with repetition, the largest possible number of matches among four teams is $\frac{C(4, \lfloor \frac{4}{2} \rfloor)}{2^4}... | 16 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | [
"First, I need to simplify the given expression by combining the fractions. The expression is \\(\\frac{1}{1+x+xy} + \\frac{1}{1+y+yz} + \\frac{1}{1+z+zx}\\).",
"I notice that each denominator has a common factor of \\(1 + z + zx\\). I can factor this out to simplify the expression.",
"After factoring, I combin... | The expression simplifies to $\frac{1}{z(x+y+z)}$. Expanding and using $z(x+y+z) = z^2$ yields $\frac{z^2}{z^2} = 1$. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | [
"To solve the expression \\( \\frac{1}{1+x+xy} + \\frac{1}{1+y+yz} + \\frac{1}{1+z+zx} \\), follow these steps:",
"1. **Combine the fractions:**",
"\\[",
"\\frac{1}{1+x+xy} + \\frac{1}{1+y+yz} + \\frac{1}{1+z+zx} = \\frac{1}{1+x+y+z}",
"\\]",
"2. **Simplify the fraction:**",
"\\[",
"\\frac{1}{1+x+y+z}... | The expression simplifies to $\frac{1}{x+y+z}$ by combining fractions and using the given condition $xyz=1$. Substituting $x+y+z=1$ into the simplified fraction results in $1$. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | [
"First, I need to simplify the given expression by combining the fractions with the same denominator. This allows me to combine the fractions into a single fraction.",
"Next, I recognize that each fraction has a common denominator, so I can add or subtract the numerators directly.",
"After simplifying the expre... | The expression simplifies to $\frac{x+y+z}{(x+y+xz)(y+z+xy)}$, which equals $1$ for all real $x,y,z$ provided $\text{x} \cdot \text{y} = 1$. Thus, the value of the expression is $1$. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | [
"First, I observe that the given equation involves adding three fractions with the same denominator: \\( \\frac{1}{1+x+xy} + \\frac{1}{1+y+yz} + \\frac{1}{1+z+zx} \\).",
"I notice that each fraction has \\( x + y + z \\) in the denominator, and all the numerators add up to 1. This suggests that the denominators o... | Since the numerators sum to 1 and the denominators sum to 1+z+xyz=1+zyz, the left side must equal 1. With xyz = 1, we get x + y = 1/5 and z + zy = 1/x + xz, giving 1/(x+y+5) = 1/x+7. | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | 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? | [
"First, I need to find the root(s) of each equation \\( f(x) = 0 \\) and \\( g(x) = 0 \\).",
"For \\( f(x) = x^3 - 3x + b \\) with \\( f(x) = 0 \\):",
"1. Set the equation equal to zero:",
"\\[",
"x^3 - 3x + b = 0",
"\\]",
"2. Factor out \\( x \\) from the polynomial:",
"\\[",
"x(x^2 - 3x + b) = 0",... | The possible values of \( b \) are \( 2 \) and \( -2 \). The sum of these values is \( 2 + (-2) = 0 \). | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | 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? | [
"First, set each function equal to zero:",
"1. \\( f(x) = 0 \\) gives \\( x^3 - 3x + b = 0 \\)",
"2. \\( g(x) = 0 \\) gives \\( x^2 + bx - 3 = 0 \\)",
"For the quadratic equation \\( x^2 + bx - 3 = 0 \\) to have real solutions, its discriminant must be non-negative:",
"\\[ b^2 + 16 \\geq 0 \\]",
"\\[ b^2 ... | The sum of possible values of \( b \) such that the differential equation \( \frac{d}{dx}(f(x)) = 0 \) and the linear equation \( g(x) = 0 \) have a common root is 0, provided \( b^2+16 \geq 0 \), which holds true for \( b \in \{8, -8\} \). Therefore, \( 8 + (-8) = 0 \). | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | 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? | [
"To solve the problem, we start by setting the two polynomials equal to zero:",
"1. \\( f(x) = 0 \\)",
"2. \\( g(x) = 0 \\)",
"For the equation \\( f(x) = 0 \\) to have exactly one root, the constant term must be zero, which means \\( b = 0 \\).",
"Next, we verify the other equation \\( g(x) = 0 \\):",
"1... | Setting \(f(x)=0\) and \(g(x)=0\) gives \(b=0\), and since \(x-b=0\) only at \(b=0\), the only roots are \(x=-3\) and \(x=0\). Thus the sum of all possible values of \(b\) is \(\boxed{0}\). | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that $(a_n)$ is a sequence of real numbers such that the series $$ \sum_{n=1}^\infty\frac{a_n}n $$ is convergent. Show that the sequence $$ b_n=\frac1n\sum^n_{j=1}a_j $$ is convergent and find its limit. | [
"Okay, let me try to figure out this problem. So, we have sequences where the series ∑_{n=1}^∞ a_n is convergent, and we need to prove that the sequence b_n = 1/n·∑_{j=1}^n a_j is convergent and find its limit. Hmm, I remember that Cauchy sequences are convergent. Wait, but first I should see if the above series co... | The limit of \(b_n = \frac{1}{n}\sum_{j=1}^n a_j\) is zero by Cauchy analysis, since harmonic series convergence applies directly to these terms (taking the limit as \(a_1 = 1\) implies \(\lim_{k\to 1} a_k = \lim_{k\to 1} \frac{a_1^k}{n} = \frac{1}{0}\)).
\boxed{0} | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$ .
*Proposed by Eugene Chen* | [
"To find the prime \\( p \\) such that \\( f(p) = 418{,}195{,}493 \\) for the function \\( f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5 \\), we start by setting up the equation:",
"\\[",
"(p^4 + 2p^3 + 4p^2 + 2p + 1)^5 = 418{,}195{,}493",
"\\]",
"First, we multiply both sides by \\( (p^5 + 1)^5 \\) to eliminate th... | The equation simplifies to \( p^4 + 2p^3 + 4p^2 + 2p + 1 = 418{,}195{,}493 \pmod{(p^4 + 1)^5} \), which implies \( p^4 + 1 \) is constant. Testing prime factors, we find \( 418{,}195{,}493 = 2^4 + 1 \), yielding \( p=2 \) as the prime factor. | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$ .
*Proposed by Eugene Chen* | [
"To solve for the prime \\( p \\) such that \\( f(p) = 418{,}195{,}493 \\) for the function \\( f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5 \\), we set up the equation:",
"\\[",
"f(p) = p^5 + p^4 + p^3 + p^2 + p + 1 = 418{,}195{,}493",
"\\]",
"Computing the expansion:",
"\\[",
"p^5 + p^4 + p^3 + p^2 + p + 1 =... | To find the prime \( p \), we set \( f(p) = p^5 + p^4 + p^3 + p^2 + p + 1 = 418{,}195{,}493 \) and solve \( xx^5 + x^4 + x^3 + x^2 + x + 1 = x^8 + x^7 + x^5 + x^3 \). The prime root of \( 418{,}195{,}493 \) is \( 2^{5} \), so \( p=2 \). | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$ .
*Proposed by Eugene Chen* | [
"To find the prime factor of \\( f(p) \\), we start by setting up the polynomial equation:",
"\\[",
"f(p) = (p^5 + p^4 + p^3 + p^2 + p + 1)^5 = 418,195,493",
"\\]",
"We can rewrite this equation as:",
"\\[",
"(p^5 + p^4 + p^3 + p^2 + p + 1) \\cdot (p^4 + 1) \\cdot (p^3 + p) = 418,195,493",
"\\]",
"N... | The prime factor \( p \) is found to be 3 after completing the factorization of the generated polynomial \( (p^5 + p^4 + p^3 + p + 1)(p(p + 1)^2)(p) \) and substituting \( p=3 \) into the factorization of \( p(p + 1)^2 \). The prime factor associated with \( p=3 \) is \( 2 \). | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares. | [
"First, I need to determine the total number of unit squares in the \\( 9 \\times 9 \\) square. Since each \\( 2 \\times 3 \\) rectangle contains 2 black and 4 white unit squares, and there are 9 \\( 2 \\times 3 \\) rectangles, the total number of black unit squares is calculated by multiplying the number of rectan... | The total number of unit squares is $9 \times 9 = 81$. The number of white unit squares is $9 \times 3 \times 2 = 54$, so the number of black unit squares is $81 - 54 = 27$. | 27 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$ . | [
"To determine the values of \\( m \\) and \\( n \\), we analyze the number \\( 2^{1997} \\) and \\( 5^{1997} \\) individually and collectively.",
"1. **Terminal Versus Terminal:**",
"- The number \\( 2^{1997} \\) ends after 1997 digits.",
"- The number \\( 5^{1997} \\) ends after 1997 times 5, which is \\( 29... | The number of terminal digits in \( 2^{1997} \) is \( \lfloor \log_2{1997} \rfloor = 4 \), so \( m = 4 \). The number of terminal digits in \( 5^{1997} \) is \( \lfloor 1997 \rfloor = 1996 \), so \( n = 1996 \), resulting in \( m + n = 4 + 1996 = 1998 \). | 1998 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$ If no element of $M$ is equal to the sum of any two elements of $M,$ find the least value that the greatest element of $M$ can take. | [
"To determine the least value that the greatest element of the set \\( M \\) can take, we start by noting that \\( M \\subset \\mathbb{N}^* \\) has 2004 elements.",
"Since \\( |M| = 2004 \\), the greatest element of \\( M \\) must be one less than this value. Therefore, the greatest element \\( X \\) is \\( 2004 ... | The greatest element of \( M \subset \mathbb{N}^* \) such that no element is equal to the sum of any two elements is \( M_2 = 2004 - 1 = 2003 \). The least value the greatest element can take is the sum of the two smallest elements itself, which is \( 2003 + 2004 = 4007 \). | 4007 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$ . | [
"First, I need to determine the number of permutations of the letters \\( ABCDE \\) under the given constraints.",
"Since \\( A \\) and \\( B \\) cannot be adjacent, I'll calculate the permutations of the remaining 3 letters \\( CDE \\) and 2 letters \\( AB \\). These letters form a part of one of the edges of a ... | Given ABCDE as the set of letters, if A and B are not adjacent and C and D are not adjacent, the valid permutations are AB' (2 P), CD' (2 P), AC' (2 P), AD' (2 P), AE' (1 P), and BE' (1 P). There are a total of 48 such permutations. | 48 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$ subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$ , then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$ . | [
"Okay, so I have this problem here about a set M that's 30 elements long. It's divided into subsets, and there's a condition: the sum of two elements a and b equals n squared, where n is an integer. Also, a and b have to be different, and this sum n squared must be an integer. I need to find the minimum value of k.... | The minimum value of k is 3. This is because for each element x of the set, x contributes 3 to the sum of y+x. | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Alice is counting up by fives, starting with the number $3$ . Meanwhile, Bob is counting down by fours, starting with the number $2021$ . How many numbers between $3$ and $2021$ , inclusive, are counted by both Alice and Bob? | [
"First, I need to identify the numbers that are divisible by both five and four within the range of 3 to 2021.",
"To do this, I'll find the least common multiple (LCM) of 5 and 4, which is 20. This means I need to count all multiples of 20 that lie between 3 and 2021.",
"Next, I'll divide 2021 by 20 to determin... | The numbers divisible by both 5 and 4 are those that are multiples of their LCM, which is 20. There are 101 multiples of 20 between 3 and 2021 inclusive, resulting in 101 numbers counted by both Alice and Bob. | 101 | true |
Open-Reasoner-Zero/orz_math_57k_collection | What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours? | [
"To determine the minimum number of colours needed to color all positive real numbers such that two numbers with a ratio of 4 or 8 have different colours, follow these steps:",
"1. **Understand the Ratio Conditions:**",
"- For two numbers to have the same ratio (4r or 8r), the ratio of their absolute values mus... | The numbers with a ratio of 4 or 8 must have the same or opposite absolute values of \( r \). Numbers with characteristic values of 2 (\( r=2 \)) and 4 (\( r=4 \)) meet this criterion, resulting in 3 colours (red and blue). | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Determine all positive integers $n$ such that $$ n\cdot 2^{n-1}+1 $$ is a perfect square. | [
"To determine the positive integers \\( n \\) such that \\( n \\cdot 2^{n-1} + 1 \\) is a perfect square, we first observe the expression. The expression can be factored as \\( (n-1) \\cdot 2^{n-1} + 1 \\). For this to be a perfect square, the exponent \\( n-1 \\) must be a perfect square.",
"Let \\( n-1 = k^2 \\... | For \(n\cdot 2^{n-1}+1\) to be a perfect square, \(n\) must be of the form \(2k+1\), which gives \(2^{4k+4}\), so \(2^{4k+4}\) is a perfect square. For example, \(n=1, k=1\): \(1(2+1)^2+1 = 5\), while \(n=2, k=0\): \(2^1+1=5\) (with no perfect squares). | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Determine all positive integers $n$ such that $$ n\cdot 2^{n-1}+1 $$ is a perfect square. | [
"First, I need to determine all positive integers \\( n \\) such that the expression \\( n \\cdot 2^{n-1} + 1 \\) is a perfect square.",
"To do this, I should factor the expression into a perfect square and a constant term.",
"Looking at the exponent part, \\( 2^{n-1} \\), I observe that \\( 2^{n-1} \\) can be ... | The expression \( n \cdot 2^{n-1} + 1 \) can be rewritten as \( \left( \frac{1}{2} \right)^n + 1 \), which is \( \left( \frac{a}{b} \right)^2 + 1 \) where \( a = \frac{1}{2} \) and \( b = 2 \). Therefore, \( n=5 \). | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = 2^x + 3^x$ . For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$ ? | [
"First, I need to determine the prime numbers \\( p \\) such that \\( 2^p \\leq f(n) < 3^p \\) for all \\( n \\) from 1 to 2020.",
"The function \\( f(x) = 2^x + 3^x \\) can be rewritten as \\( f(x) = 2^x \\pmod{3} + 3^x \\pmod{2} \\).",
"To find the condition for \\( f(x) \\) to be relatively prime to \\( 2^p ... | For \(n \in \{1, \dots, 2020\}\), \(f(n)\equiv2\pmod{2}\), while only \(p=4\) (with \(2^p\equiv1+3\equiv1\pmod{2}\)) causing unsatisfying \(2^p\equiv1+3\equiv1\pmod{2}\) and thus independence of \(2^p\) from \(2^p\). The primes satisfying these conditions only are \(4, 5, \dots, 11,\dots\), so there are **11** such int... | 11 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$ . | [
"Let \\( A \\) and \\( B \\) be distinct positive integers. Both have the same number of positive divisors as 2013.",
"**Step 1: Find the number of divisors of 2013.**",
"The number of divisors of a number can be determined by its prime factorization. Let\\( 2013 = p^2 \\cdot q^3 \\), where \\( p \\) and \\( q ... | The number of divisors of \( 2013 = p^2 \cdot q^3 \) is 12, so \( |A - B| \leq 12 \) with \( A = 2013 \). The smallest positive value for \( B \) that also satisfies \( 1 \leq A \leq \infty \) and \( 1 \leq B \leq \infty \) is \( 2013 - 2012 = 1 \). | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$ . | [
"To find the least possible value of \\( |A - B| \\) where \\( A \\) and \\( B \\) are distinct positive integers, we start by noting that 2013 is a prime number. Its only positive divisors are 1 and 2013.",
"Since \\( A \\) and \\( B \\) must share the same number of divisors as 2013, they must be distinct prime... | Since 2013 has exactly 2 divisors it must be a prime like 2 and 3, giving \(|A - B| = |2 - 3| = 1\) for distinct primes. The least possible value is \(\boxed{1}\). | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime? | [
"To determine the minimum number of numbers that need to be deleted to ensure that the sum of any two selected numbers is not a prime, we can follow these steps:",
"1. **List the numbers from 1 to 50:**",
"\\[",
"1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27... | The minimum number of elements that need to be deleted to ensure that the sum of any two selected numbers is not a prime is 4; deleting the pairs \((29, 31)\) and \((31, 43)\) reduces the count of prime numbers from 35 to 29. This leads to a final set of 25 numbers. | 25 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I have this problem where I need to show that the sum of the sides of a convex polygon P that's inside a square of side length 1 is less than or equal to 4. Hmm, let me think about how to approach this.",
"First, I remember that when you have a square and a convex polygon, as long as the polygon is conv... | Suppose a convex polygon is inscribed in a square of side length 1 rotating in a way that conserves the perimeter through polygon rotation. By carefully calculating this rotation, we find the sum of the sides of the polygon is less than or equal to 4. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I need to show that the sum of the sides of any convex polygon P that's contained in a square with a side length of 1 is less than or equal to 4. Hmm, let me think about how to approach this.",
"First, I know that the square is a two-dimensional shape with all sides equal in length. Now, a convex polygo... | The sum of all sides of a convex polygon P contained in a square of side length 1 is equal to the number of vertices P, which is $V=4$. This was shown as Euler's Formula $V - E + 0 = 2$, which simplifies to $V - E = 2$, and applying this in the case with P on the square's vertices yields $V = 4$. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I need to show that the sum of the sides of any convex polygon P contained in a square with side length 1 is less than or equal to 4. Hmm, let's break this down.",
"First, let me recall that the Euler characteristic of a convex polygon with vertices V, edges E, and sides s is Γ = V - E + s. Since the sq... | The sum of the sides of \( P \) is less than or equal to 4 because the polygon \( P \) is contained within a square with side length 1. This is because the sum of horizontal edges (vertically changing segments) is always 2 over the number of sides, but the number of vertical edges changes based on the horizontal edges,... | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I need to show that the sum of the sides of a convex polygon P inside a square equals 4 or more. Hmm, let's start by recalling some properties of quadrilaterals in a square.",
"First, a square has four sides of equal length. But we're considering a convex polygon P that we don't have to worry about any ... | The sum of any two sides of a convex polygon inside a square must be greater than or equal to the third side. Therefore, it must be at least 4. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I have this problem where a convex polygon P is contained within a square of side length 1. I need to show that the sum of the sides of P is less than or equal to 4. Hmm, let me try to visualize this first.",
"Alright, so there's a square here, and inside it, we have a convex polygon. The square has sid... | The sum of the sides of the convex polygon \( P \) is at least \( 4 \). Thus, the maximum sum of its sides is \( 4 \). | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"First, since \\( P \\) is contained within a square with side length 1, it can be expressed as a smaller square inside the larger square. Each side of \\( P \\) is therefore adjacent to one side of the square, making it longer than one side of the square.",
"Next, let's determine the maximum possible size of a s... | Since \( P \) is inside a square of side length 1, its inner square must have side length 2. Hence the sum of its sides is greater than or equal to 4. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Okay, so I have this problem where a convex polygon P is contained in a square with side length 1. I need to show that the sum of the sides of P is less than or equal to 4. Hmm, interesting. Let me think about how to approach this.",
"First, let me visualize the square. It has a side length of 1, so if I imagine... | A polygon with side length 1 can only contain rectangles, parallelograms, or triangles, all of which have side lengths strictly between 1 and 1. Therefore, the sum of the sides of P is strictly greater than or equal to 4, and the minimum possible sum is 4. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | [
"Alright, so I need to prove that the sum of the sides of a convex polygon P contained within a square of side length 1 is less than or equal to 4. Hmm. Okay, let's start by recalling what a square is. A square is a four-sided figure, and it's flat, meaning all its sides are the same length. The problem is consider... | The sum of the sides of P, denoted as \(S\), is less than or equal to 4. This is derived from focusing on lengths beyond a given distance from the square, using the concept of an infinitely thin polygon undergoing an infinity of moves away from the square as the vertices from its edge shrink. | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | In triangle $ABC$ , $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$ . Find $CE$ given that $AC=35$ , $BC=7$ , and $BE=5$ . | [
"First, I need to determine the measure of angle \\( ABD \\). Since point \\( D \\) lies on side \\( AC \\) and \\( \\angle ABD \\) is right, it means \\( \\angle ABD = 90^\\circ \\). Knowing that the sum of angles in a triangle is \\( 180^\\circ \\), I can calculate \\( \\angle ABD \\) as follows:",
"\\[",
"\\... | In triangle \(ABC\), \( \angle ABC \) is obtuse, and point \(D\) with angle bisector \( BE \) lies on \( AC \), such that \( \angle ABD = 105^\circ \). Given \( BE = 5 \), \( BE + CE = 10 \) is derived using the Law of Cosines, leading to \( CE = 10 \). | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | [
"Okay, so I have this problem about a regular octagon with sides of 10 units. The circles with centers A and D intersect at point I, and I need to find the length of IF. Hmm, let me think about how to approach this.",
"First, I remember that for any circle with center I, the points I, C, and A lie on a straight l... | Since \(I\) lies on the circle with radius 10 and on both circles, it is equidistant from \(A\) and \(C\), making \(IF = 10\). Note: \(\boxed{10}\). | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | [
"First, I need to determine the distance between points \\( C \\) and \\( I \\) on the circle centered at \\( A \\). Since \\( AC \\) is twice the radius \\( AC \\), the length \\( IF \\) is half the distance from \\( C \\) to \\( I \\).",
"Next, I'll use the Pythagorean theorem to find the distance between point... | The distance from \( C \) to \( I \) is \( IF = \frac{AD}{2} \). Using the Pythagorean theorem with \( BC = AD = 6 \) and \( AC = 10 \), we find \( IF^2 = (AD/2)^2 + (AC/2)^2 = 9 + 1 = 10 \). | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | [
"First, I recognize that the circle with radius \\( AC\\) is the circumcircle of the regular octagon \\(ABCDEFGH\\), meaning it is centered at \\(A\\) and its radius is 10 units. The intersection point \\(I\\) where this circle intersects another circle with radius \\(BC\\) (which is side length 10) is given as poi... | The length of segment \( IF \) is \( 10 \). This is derived by observing that \( \triangle BIB \) and \( \triangle CAB \) are similar, \( \frac{IW}{10} = \frac{10}{10} \), so \( IF=10 \). | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | [
"First, I recognize that a regular octagon has all its sides of equal length and all its internal angles equal. Given that each side of the octagon is 10 units long, the radius \\( AC \\) is also 10 units.",
"Next, the circle centered at \\( A \\) with radius \\( AC \\) intersects the circle centered at \\( D \\)... | The radius of the circle with center \( D \) is equal to the radius of the circle with center \( A \), which is 10 units. Therefore, the length of segment \( IF \) is \( 10 \) units. | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $O$ be a circle with diameter $AB = 2$ . Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$ , and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ ca... | [
"Alright, so I have this problem about circles, and I need to find the minimum possible area of two circles O₁ and O₂ tangent to each other with the same diameter AB. The condition is that O₁ is tangent to O₂ at A and to O₃ at B, meaning O₁ is external to O₂ and external to O₃, respectively. Also, O₁ and O₂ are ext... | The minimum possible area is 3π. This is found by setting the minimum equality in the area formula to π, thus resulting in a minimum area of 3π. | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$ ? | [
"First, I need to solve the equation \\(2abc = a + b + c + 4\\) for all possible integer values of \\(a\\), \\(b\\), and \\(c\\).",
"To simplify the equation, I'll divide each term by 2:",
"\\[",
"abc = a + b + c + 2",
"\\]",
"Next, I notice that the equation can be rewritten as:",
"\\[",
"abc(a + b +... | Dividing the equation by 2 gives \(abc = a + b + c + 2\). Simplifying, we get \(abc(a + b + c) = 2(a + b + c) + 2\), which implies that \(abc\) depends on each pair \((a,b,c)\), resulting in 6 ordered triples. | 6 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the least positive integer $n$ such that $15$ divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers $a_1, a_2, \dots, a_{15}$ . | [
"To determine the least positive integer \\( n \\) such that \\( 15 \\) divides the product \\( a_1 \\dots a_{15} (a_1^n + a_2^n + \\dots + a_{15}^n) \\), we start by expressing the product in terms of a single factor:",
"\\[",
"(a_1 + a_2 + \\dots + a_{15})(a_1^n + a_2^n + \\dots + a_{15}^n)",
"\\]",
"This... | The product \( (a_1 + a_2 + \dots + a_{15})(a_1^n + a_2^n + \dots + a_{15}^n) \) requires at least three factors of \( 1^n \) to guarantee \( 15 \) divides it. The least positive integer \( n \) satisfying this condition is \( 4 \). | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ *Proposed by Ray Li* | [
"Okay, so I have this geometry problem. Let me try to visualize it. There's a triangle ABC with area 5, and side BC is 10 units long. Euclid's parallel progress theorem says that if one of the diagonals of a triangle splits it into two triangles with equal areas and trapezoids with the same areas on each side, then... | The area of triangle ABC is 5, so AC is 10 and AB squared + BC squared is 200. Since cutting it into two triangles with equal areas on the diagonals and matching the trapezoidal areas gives AEGF a parallelogram with area 5, the value of AB² + AC² is 200. | 200 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the smallest exact square with last digit not $0$ , such that after deleting its last two digits we shall obtain another exact square. | [
"First, I need to identify the smallest exact square with a last digit that is not 0. A number is a perfect square if it is the square of an integer, meaning it must have even numbers of 2s and 5s in its binary representation. Since '0' is the last digit not allowed, the last digit must be odd. The smallest odd dig... | The smallest exact perfect square ending in 1 where the last two digits remain the same after deletion is \(1011_2 = 11\). This binary number's decimal equivalent is 22, which is \(11^2\) and the last two digits are still present. | 121 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$ . Let $H$ and $G$ be the midpoints of $BD$ and $AC$ , respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$ . | [
"First, I need to understand the problem. We have a convex quadrilateral ABCD, and we're given points D and B, extended to points E and C respectively. The midpoints H and G are located at positions B and A.",
"Next, I'll identify the positions of the midpoints:",
"- Midpoint H at position B",
"- Midpoint G a... | The area of triangle \( EHG \) is approximately \( \frac{1}{4} \) of the total area of the quadrilateral \( ABCD \), if \( AB = AD = BC \). This is because the trapezoid formed by \( \triangle EHG \) and \( \triangle ABCD \) shares the same vertical height as \( \triangle ABCD \), making its area equal to \( \frac{1}{4... | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$ . Let $H$ and $G$ be the midpoints of $BD$ and $AC$ , respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$ . | [
"Alright, so I have this geometry problem here about a convex quadrilateral ABCD. The sides AD and BC are extended to meet at point E. Also, H and G are the midpoints of BD and AC respectively. I need to find the ratio of the area of triangle EGH to that of the quadrilateral ABCD.",
"Hmm, let me visualize this. M... | The area of triangle EGH is 1/4 of the area of the quadrilateral ABCD. | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$ . Let $H$ and $G$ be the midpoints of $BD$ and $AC$ , respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$ . | [
"First, I need to determine the ratio of the areas of triangles \\( EHG \\) and \\( ABCD \\).",
"I know that the midpoint \\( H \\) is located at \\( E \\), so \\( E \\) has the property that \\( E \\cdot H = BD \\cdot H \\). Similarly, \\( H \\) is the midpoint of \\( AC \\), meaning \\( H \\cdot G = AC \\cdot H... | With \(E\) and \(H\) the midpoints of \(BD\) and \(AC\) giving \(E\cdot H = 2B\) and \(H\cdot G = AC\cdot H = 2C\); the area of \(EHG\) is half of \(E \cdot H = 2B\) while the area of \(ABCD\) is \(BD \cdot AC = 2BCD = 2BC \cdot 2B = 4BCD\). Thus \(\frac{\text{Area}(EHG)}{\text{Area}(ABCD)} = \frac{B}{4C \cdot B} = \bo... | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | Given an integer $n \geq 2$ determine the integral part of the number $ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$ | [
"Okay, so I need to find the integral part of this sum of reciprocals from n down to 2, and then from 2 down to n. Both sums are alternating sums of reciprocals. Hmm, how do I approach this?",
"First, for the first sum, the n from 1 to n-1, it should include every term from 1 up to n-1. So that's n-1 + 1 + ... + ... | The integral part of the first sum \( \dbinom{n-1}{n} + 1 + \cdots + 1 \) is 1. The integral part of the second sum \( \sum_{k=2}^{n-1} \frac{2}{k} + 2 \) is 0 because it is evaluated from 2 to 1, and its average is \( \frac{1}{2} - \frac{1}{2^2} = 0 \). | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos? | [
"First, determine the number of ways to choose one wrap out of five for each type of burrito: 5 choices for beef burritos and 5 choices for chicken burritos.",
"Calculate the number of ways to choose one wrap for all beef burritos: 5 combinations.",
"Similarly, calculate the number of ways to choose one wrap fo... | Spencer can make 25 different burritos by selecting one of 5 wraps for each beef or chicken burrito type. This is calculated as 5 combinations for beef burritos multiplied by 5 combinations for chicken burritos, resulting in 25 total combinations. | 25 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos? | [
"First, I need to determine the number of ways Spencer can arrange the two types of burritos, beef and chicken, using the five available wraps. Since the deciding factor is the use of wraps, I focus on permutations of beef and chicken burritos.",
"For burritos made with beef, there are \\(5!\\) possible orders. S... | Since the deciding factor is wraps, the total number of arrangements is \(5! \times 5! = 25\). Therefore, Spencer can make **25** distinct orders for the five burritos. | 25 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | [
"First, I need to determine the possible values of \\(4^a + 2^b\\) where \\(a\\) and \\(b\\) are nonnegative integers and \\(4^a + 2^b\\) is a perfect square. A perfect square can be expressed as \\((\\sqrt{4})^2 = 2^2 = 4\\), so \\(4^a + 2^b\\) must be a sum of two perfect squares.",
"Next, I analyze each possib... | The possible sums \( a + b \) for nonnegative integers \( a \), \( b \), such that \( 4^a + 2^b + 5 \) is a perfect square are 1 and 8. The sum of these valid sums is \( 1 + 8 = 9 \). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq 2$ . If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$ ? $
\textbf{a)}\ 18
\qquad\textbf{b)}\ 19
\qquad\textbf{c)}\ 28
\qquad\textbf{d)}\ 29
\qquad\textbf{e)}\ \te... | [
"To find the least possible value of \\( N \\) such that each of the numbers from 1 to 9 is in the same circular arrangement, we start by noting that each number shares a common digit with its two immediate neighbors in decimal.",
"1. **Analyzing the first number (1):**",
"- The digits of 1 are 1.",
"- Since ... | Each number shares a common digit with its two immediate neighbors, and keeping the sum of digits divisible by three forces the digits to multiply to give a product that's 3 modulo 3. Thus the numbers lie together in larger circular groups; checking for 2, 3, and 4 reveals solutions in either 2 or 4, giving the least p... | 29 | true |
Open-Reasoner-Zero/orz_math_57k_collection | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | [
"Okay, so I need to figure out the greatest possible number of steps C required to make black all 8x8 square, which is a 64 unit square. Let me start by visualizing the grid. There are 8 rows and 8 columns, making 64 cells in total.",
"First, I know that in each of the cell (the 1x1 rectangle) black squares are e... | The total number of unit squares is 8 × 8 = 64, and the average is 32/2 = 16. For 64, 32 ≤ 64 ≤ 48, so the optimal minimum is 32. | 32 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"To determine \\( f(9) \\), we start by analyzing the given functional equation \\( f(x + y) = f(xy) \\) for all \\( x \\geq 4 \\) and \\( y \\geq 4 \\).",
"First, we observe that the equation implies \\( f(x + 8) = f(8y) \\). This suggests a relationship similar to the cubic equation \\( x^3 = y \\), which has a... | By computing the right side of the functional equation \(f(x+y) = f(xy)\) for \(x\ge 4\) and \(y\ge 4\), we observe a pattern that extends from \(f(8)=9\). Thus, \(f(9)=9\). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"Okay, so I have this problem where I need to find the value of f(9) given that f(x + y) equals f(xy) for all positive integers x and y. They also tell me that f(8) equals 9. Hmm, let me try to break this down step by step.",
"First, let me understand the given condition. For any two positive integers x and y, f(... | Since \( f(8) = 9 \), we know \( f(3) = 9 \) because \( f(4) = 8 \) applies, and for \( f(4) > 1 \), the only integer \( A \) in the range \( \{1, 2, ..., 9\} \) such that \( f(A) \leq 9 \) is found in the interval \( (1, 4) \). Thus, \( f(9) = 9 \). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"First, I need to understand the given functional equation \\( f(x \\times y) = f(x) \\times f(y) \\) for all positive integers \\( x \\ge 4 \\) and \\( y \\ge 4 \\). This means that if I know the value of \\( f \\) at \\( x \\), I can find \\( f \\) at \\( x \\times y \\) by multiplying the two values together.",
... | The generalized Fibonacci sequence implies \( f(32) = f(28) \), which is confirmed by the recurrence \( f(n+2) = f(n+1) \) for \( n=4 \). Since \( 16 = 8+4 \), \( f(16) = f(8) = 9 \), and therefore \( f(9) = f(12) = 9 \). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"To determine \\( f(9) \\) given the functional equation \\( f(x + y) = f(xy) \\) and the condition \\( f(8) = 9 \\), I start by evaluating \\( f(8) \\).",
"First, since \\( 8 \\geq 4 \\), I can express \\( f(8) \\) as:",
"\\[",
"f(8) = f(8 \\times 8) = f(64)",
"\\]",
"Next, substituting back \\( 64 \\) f... | Given \( f(x + y) = f(xy) \) and \( f(8) = 9 \), we find \( f(8) = f(64) = f(4096) \). As \( f(8) = 9 \), and \( f(4096) = 9 \), we conclude that \( f(9) = f(7) \), thus \( f(9) = 9 \). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"Okay, so I have this problem here where I need to find f(9). Let me try to understand what's given and what I need to find.",
"The sets X and Y are defined as X = {8, 9, 10, ..., x-1} and Y = {8, 9, 10, ..., y-1} for some positive integers x and y. There's a function f: X -> Y such that f(x + y) = f(xy) for all ... | The problem requires us to find \( f(9) \). Given \( f(x+y) = f(xy) \) with \( x \geq 4 \) and \( y \geq 4 \), and \( f(8) = f(9) \), we know that \( f(8) \) can only be 9. Substituting \( y=4 \) into \( f(9) \) and satisfying \( 4 \leq y \leq 12 \), we find \( y=4 \), allowing \( y=5 \) (as \( x=9 \) must be at least ... | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"First, I need to understand the given function \\( f: X \\rightarrow X \\) that satisfies the functional equation \\( f(x + y) = f(x \\times y) \\).",
"Since \\( f \\) is defined for all \\( x \\geq 4 \\) and \\( f(8) = 9 \\), I can use the functional equation to verify the function's properties.",
"I'll start... | Given that \( f(x + y) = f(xy) \) for \( x \geq 4 \) and \( f(8) = 9 \), the functions satisfy the functional equation. Since \( f(4) = 9 \) and \( f(8) = f(4 \times 4) = f(16) \), it follows that \( f(9) = f(8) = 9 \). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$ | [
"Alright, so I have this problem here about functions and integers. Let me try to understand what it's asking. There's this set X of positive integers greater than or equal to 8. So, X would be... let me list out some numbers: 8, 16, 24, 32, and so on. That's huge!",
"Now, the function f: X → X is defined such th... | Given that \(f(x + y) = f(xy)\) for \(x \ge 4\) and \(f(8) = 9\), it follows that for \(x = 8\), \(y = 12\). Thus, \(f(9) = f(64) = 9\). | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a$ , $b$ , and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$ . Over all possible values of $a$ , $b$ , and $c$ , determine the maximum possible value of $a + 2b + 3c$ .
*Proposed by Andrew Wen* | [
"First, we are given the equation \\( 2a + b + c = 10 \\) and need to find the maximum value of \\( a + 2b + 3c \\) over all possible values of \\( a, b, c \\) within the interval \\( 0 \\leq a, b, c \\leq 5 \\).",
"To maximize \\( a + 2b + 3c \\), we can express \\( 2a + b + c \\) in terms of \\( b + c \\):",
... | The maximum value of \( a + 2b + 3c \) is maximized when \( a, b, c \) are integers such that \( a + b = 5 \) and \( c = 10 - a \), which yields \( a = -5 \). Substituting these values into \( 2a + b + c = 10 \) gives \( 2a + 10 - 2a + c = 10 \), resulting in a maximum of \( 25 \). | 25 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes. | [
"To find the sum of all integers \\( n \\) for which \\( n - 3 \\) and \\( n^2 + 4 \\) are perfect cubes, we start by recognizing that both expressions must be of the form \\( k^3 \\), where \\( k \\) is an integer.",
"First, we solve for \\( k \\) in the equation \\( n - 3 = k^3 \\):",
"\\[",
"n = k^3 + 3",
... | The integers \( n \) satisfying \( n=k^3+3 \) and \( n^2+4=k^3 \) are found by pairing \( k^3 \) with \( k^3-4 \), yielding \( 6 \) and \( 7 \). Their sum is \( 6 + 7 = 13 \). | 13 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$ .
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$ .
(ii) Determine the sma... | [
"Okay, so I need to figure out how to show that the real-valued function f(n+4a) equals f(n) for all positive integers n. That seems like a kind of generalization of the pigeonhole principle, right? If I have n + 4a elements, each of which can be sorted in n ways. That should cover all the possible categories, one ... | For \(f(n+4a)=f(n)\) for all positive integers \(n\), avoid dividing by “negative terms”, so the identity holds for \(n+4a = 4a+1\). Quick verification for \(a=2\) confirms that \(f(n+4a)=\frac{187.5}{4a}\) yields the extreme value theorem partial solutions \(3\) and \(3\frac{7}{4}\) (with \(\frac{187.5}{4a \cdot 3}=\f... | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Given triangle $ABC$ , let $D$ , $E$ , $F$ be the midpoints of $BC$ , $AC$ , $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$ , how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral? | [
"Okay, so I have this problem about triangle ABC. D, E, F are the midpoints of sides BC, AC, and AB respectively. The problem says G is the centroid of triangle ABC, and for each angle BAC, I need to figure out how many non-similar triangles G can form where A, E, F, G divides side AB in various ways.",
"Alright,... | The cyclic quadrilateral GEFA is congruent to triangle ABC after linking the center of the circumscribed circle to segments creating the cyclic quadrilateral. Therefore, there are 2 congruent triangles. | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A four-digit number $n$ is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by $5$ , the result is $1434$ . For example, $1984$ is *literally 1434* because $1$ mod $5$ is $1$ , $9$ mod $5$ is $4$ , $8$ mod $5$ is $3$ , and $4$ mod $5$ is $4$ . Find the... | [
"To determine the sum of all four-digit positive integers that are \"literally 1434,\" followed by replacing each digit by its remainder when divided by 5, we need to convert each modified digit into its modulo 5 counterpart and add them together. The resulting sequence of sums is given by:",
"\\[",
"\\left\\{1... | The sum of these integers, with each modified number representing a digit divided by 5 plus 1 modulo 5, is calculated by converting each digit into its modulo 5 counterpart and summing them. The total sum is \(1 \mod 5 + 4 \mod 5 + 3 \mod 5 + 4 \mod 5 + 8 \mod 5 = 67384\). | 67384 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A four-digit number $n$ is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by $5$ , the result is $1434$ . For example, $1984$ is *literally 1434* because $1$ mod $5$ is $1$ , $9$ mod $5$ is $4$ , $8$ mod $5$ is $3$ , and $4$ mod $5$ is $4$ . Find the... | [
"First, recognize that we need to find four-digit numbers where each digit is replaced by its remainder when divided by 5 and the result is 1434.",
"1. **Finding the Remainders:**",
"- The digits that give a remainder of 1 when divided by 5 are 1, 6, and 4 (since 1 = 5 - 4).",
"- The digits that give a remain... | The problem requires finding four-digit numbers where each digit is replaced by its remainder when divided by 5 and the result is 1434. The possible numbers are 1434, 1349, 1449, 1439, 1934, 1944, 914, 934, and 941. The sum of these numbers is \(67384\). | 67384 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A four-digit number $n$ is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by $5$ , the result is $1434$ . For example, $1984$ is *literally 1434* because $1$ mod $5$ is $1$ , $9$ mod $5$ is $4$ , $8$ mod $5$ is $3$ , and $4$ mod $5$ is $4$ . Find the... | [
"To find the sum of all four-digit positive integers that are \"1434*\", we need to extend the sequence 1434 into a general four-digit number of the form abcd4, where a, b, c, and d satisfy the conditions given. Given the constraints, the possible numbers are 1984, 2198, 2392, 2496, 2698, 2894, 3098, and 3292.",
... | The four-digit positive integers represented by "1434*" are 1984, 2198, 2392, 2496, 2698, 2894, 3098, and 3292. Their sum is $16 \times (1984 + 2698 + 3092 + 3396 + 3694 + 3990 + 4298 + 4594 + 4890) = 67384$. | 67384 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$ , where $F_n$ is the Fibonacci sequence
( $F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$ )
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$ | [
"Okay, so I need to show that for the nth term of the Fibonacci sequence, V_n and V_{n+1} are the sides of a triangle where the area is 1/2. Hmm, let's start by recalling the Fibonacci sequence. The nth term is the sum of the two preceding ones.",
"Given that F_1 = F_2 = 1, then F_{n+1} = F_1 + F_2. So pretty str... | The sides of a triangle formed by the first \( n \) terms of the Fibonacci sequence are equal, and the area is \( \frac{1}{2} \). | \frac{1}{2} | true |
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