source stringclasses 7
values | question stringlengths 10 1.02k | steps listlengths 1 83 | answer stringlengths 9 3.01k | 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 recognize that triangle ABC is an equilateral triangle with all sides and angles equal. Given side lengths of 1 and 3, the hypotenuse \\( AB = 3 \\), hence the remaining leg \\( BC \\) is also 3.",
"The problem involves cutting this equilateral triangle away. Since the triangle is not an equilateral tri... | Let's solve the problem step by step.
**Given:**
- Triangle \( ABC \) has sides \( DEB = EB = 1 \).
- It is cut from the equilateral triangle of side length 3.
**Objective:**
Find the perimeter of the remaining quadrilateral.
**Step 1: Identify the Remaining Legs of Triangles**
Since triangle \( ABC \) is an equi... | 8 | true |
Open-Reasoner-Zero/orz_math_57k_collection | For each positive integer $n$, let
$a_n = \frac{(n+9)!}{(n-1)!}$
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is
$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ }... | [
"To determine the smallest positive integer \\( k \\) such that the rightmost nonzero digit of \\( a_k \\) is odd, we start with the definition of the function \\( a_k = \\frac{(n+9)!}{(n-1)!} \\). Here, each higher denominator \\( (n-1)! \\) reduces the number of terms in the factorization as rapidly as possible u... | To determine the smallest positive integer \( k \) such that the rightmost nonzero digit of \( a_k \) is odd, we can follow these steps:
1. **Understand the Function \( a_k \):**
The function \( a_k = \frac{(n+9)!}{(n-1)!} \) is defined such that each term in the factorization is a nonzero digit. The product \(... | 9 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy
(i)$f(x)\ge0$ for all $x$ in $[0, 1]$,
(ii)$f(1) = 1$,
(iii) $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that
$f(x) \le cx$
for every function ... | [
"Okay, so I have this problem here about functions. It says that each function f from the interval [0, 1] satisfies some conditions: (i) for all x in [0, 1], g(x) is greater than or equal to 0, and (ii) g(1) is equal to 1. Then it also states that for each x, its complement under f, which is [1, x], equals [0, 1]. ... | The smallest constant \( c \) such that \( f(x) \le cx \) for every \( 0 \leq x \leq 1 \) for every interval \([0, 1]\) is simply the constant \( c \).
\boxed{2} | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number o... | [
"Okay, let's see. I need to find the smallest number of elements in a set S where all the subsets X_i and X_{i+1} are mutually disjoint and their union is not S. Wait, the way the problem is defined: any two sets X_i and X_{i+1} are disjoint and their union is not S. So that means X_{i+1} and X_i are different, rig... | The set \( X_i \cap X_{i+1} \) are disjoint and their union is impossible, implying their union is not S. Therefore, \( S \) must be the complement of \( X_i \) in \( X_{i+1} \). The complement consists of 8 mutually disjoint subsets:
- \( Y_i \subset S \) (from \( X_i \cap X_{i+1} \))
- \( Z_i \subset Y_i \) (from \(... | 8 | 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... | [
"First, I need to understand the problem. We have a sequence of numbers, and each permutation must satisfy the condition \\( a_k = b_k \\) for some valid integer \\( k \\). However, the key condition is that since \\( a_k = b_k \\) different values of \\( k \\) are excluded, every possible permutation of \\( 1, 2, ... | To determine the number of distinct permutations of \(1, 2, \ldots, 2010\) corresponding to the numbers \(1, 2, \ldots, 2010\) where the numbers are mirrored, we can analyze the condition \(a_k = b_k \) for some valid integer \(k\).
### Key Inference:
- **Shift Pattern**: The shuffle (specifically for reversing the n... | 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 there are some permutations of the numbers 1, 2, ..., 2010 such that either all odd numbers up to k are the same or all even numbers up to k are the same, and the sum of any odd and an even number in a permutation is either 5 or 10. Hmm, this sounds a bit like the Ponta-Triscon theorem... | \boxed{1006} | 1006 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\triangle GEM$. | [
"The area of a square with side length 10 units is 100 square units. Therefore, the area of square AIME is also 100 square units.",
"Next, we consider triangle GEM with base and height equal to both the equal sides of square AIME. Given that this triangle has an area of 80 square units, maybe I can use this infor... | Let's solve the problem step by step.
**Given:**
- Square AIME has a side length of 10 units.
- The area of square AIME is 100 square units.
- By an altitude, the area of triangle GEM is 80 square units.
- Triangle GEM is isosceles with legs equal to both of the equal sides of square AIME.
**Let’s denote:**
- Side le... | 25 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) ... | [
"To determine \\( a_{2019} \\), we start with the given initial values \\( a_1 = 1 \\) and \\( a_2 = \\frac{3}{7} \\). We set up the recursive downward step-by-nine function:",
"\\[",
"a_n = \\frac{a_{n-2} \\cdot a_{n-1}}{2\\left(\\frac{a_{n-2} - a_n}{2}\\right)}",
"\\]",
"This recursive step-by-nine factor... | To determine \( a_{2019} \), we start with the given initial values \( a_1 = 1 \) and \( a_2 = \frac{3}{7} \). We set up the recursive downward step-by-nine function:
\[
a_n = \frac{a_{n-2} \cdot a_{n-1}}{2\left(\frac{a_{n-2} - a_n}{2}\right)}
\]
This recursive step-by-nine factorials matches with the factorial equat... | 8078 | 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 have this problem about probability. There's a sequence of positive integers where each member's value is a (a) and then there's a third member whose value is b. I need to prove that (a + b + c) is divisible by 3, which is 3 if a + b + c is divisible by 3.",
"First, I need to understand what a sequenc... | The probability that \( a + b + c \) is divisible by 3 is at least \(\boxed{\dfrac{1}{4}}\). | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let
$S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$
and
$S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$.
What is the ratio of the area of $S_2$ to the area of $S_1$?
$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$ | [
"To determine the ratio of the areas of \\( S_2 \\) to \\( S_1 \\), we start by defining the functions by their combinatorial algebraic expressions. Let \\( x \\) be an integer and \\( y \\) be any integer. Therefore:",
"- \\( S_1 = \\{ (x, y) \\mid \\log_{10}(1 + x^2 + y^2 ) \\le q + \\log_{10}(x + y) \\} \\), w... | The ratio of the area of \( S_2 \) to \( S_1 \) is \( \frac{1}{1} \).
The final answer is \(\boxed{\text{(E)}\ 102}\). | 102 | 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... | [
"To determine the ratio of the area of rectangle M to that of rectangle N, we first identify the coordinates of each rectangle.",
"Rectangle M has vertices at (0,0), (0,2), (0,3), (1,0), and (1,1), while rectangle N has vertex at (0,0) and corresponding x-coordinates as each of the same heights.",
"1. **Area of... | To determine the ratio of the area of rectangle \( M \) to that of rectangle \( N \), let's analyze each rectangle step by step.
### Rectangle \( M \)
- **Vertices:**
- \( M_1 = (0,0) \)
- \( M_2 = (0,2) \)
- \( M_3 = (0,3) \)
- \( M_4 = (1,0) \)
- \( M_5 = (1,1) \)
- **Intercepts:**
- \( a = 1 \)
- \(... | 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... | [
"Okay, so I have this problem about rectangles and their areas. Let me try to visualize it step by step.",
"First, there's a coordinate plane with three vertices: (0,0), (x,0), and (x,y). That gives me three points in the first quadrant, so the interval of the enlarged rectangle M. Rectangle M has points A(-1,0),... | The area of rectangle M is 1. The area of rectangle N is also 1. Therefore, the ratio of the areas is 1/1 = 1.
\[
\boxed{\frac{1}{1}}
\] | 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way t... | [
"To determine the greatest distance between Farmer Tim and Alex while they are walking in sinusoidal paths, we analyze their movements step by step.",
"1. **Initial Projection:**",
"- Farmer Tim starts at position \\( (0, 0) \\) and walks sinusoidal paths for \\( t \\) minutes.",
"- After \\( t \\):",
"\\["... | To determine the greatest distance between Farmer Tim and Alex while they are walking in sinusoidal paths, we analyze their movements step by step.
1. **Initial Projection:**
- Farmer Tim starts at position \( (0, 0) \) and walks sinusoidal paths for \( t \) minutes.
- After \( t \):
\[
(0 + t, 0)
... | 3\sqrt{3} | 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 given expression \\(\\frac{1}{1+x+xy} + \\frac{1}{1+y+zy} + \\frac{1}{1+z+zy}\\), we recognize that the numerator \\(\\frac{1}{1+x+xy}\\) and the denominator \\(\\frac{1}{1+y+zy}\\) involve different algebraic expressions with respect to \\(x, y, z\\).",
"1. **Identify Common Terms**:",
"- \\(\\fr... | To solve the given expression \(\frac{1}{1+x+xy} + \frac{1}{1+y+zy} + \frac{1}{1+z+zy}\), we can follow these steps:
1. **Identify Common Terms**:
- The numerator is \(\frac{1}{1+x+xy}\).
- The denominator is \(\frac{1+y+zy}\).
2. **Substitute Similar Terms**:
- Let \(k = y\), \(m = z\), and \(n = zx\). Subs... | 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 problem, I start by adding the fractions \\( \\frac{1}{1+x+xy} \\), \\( \\frac{1}{1+y+yz} \\), \\( \\frac{1}{1+z+zx} \\), and \\( \\frac{1}{1+x+xz} \\) to get \\( 0 \\). Since \\( x, y, z \\) are real numbers, the operations must result in zero.",
"I examine the coefficients \\( x \\) and \\( z \\).... | To solve the given expression, we start by combining the fractions:
\[
\frac{1}{1+x+xy} + \frac{1}{1+y+yz} + \frac{1}{1+z+zx} = 0
\]
Since \( x, y, z \) are real numbers, the resulting expression must give zero. Let's analyze the coefficients of each term:
1. **First Term:** \( \frac{1}{1+x+xy} \)
- Factor out \(... | 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 expression can be grouped as the sum of a polynomial and another polynomial adding fractions together with real numbers.",
"Iexperiment with combining the series:",
"\\[",
"\\frac{1}{1+x+xy} + \\frac{1}{1+y+yz} + \\frac{1}{1+z+zx}",
"\\]",
"I notice that all terms in the t... | To evaluate the expression \(\frac{1}{1+x+xy} + \frac{1}{1+y+yz} + \frac{1}{1+z+zx}\) given that \(xyz = 1\), we can combine the terms separately.
**Step 1: Group the Terms**
\[
\frac{1}{1+x+xy} + \frac{1}{1+y+yz} + \frac{1}{1+z+zx}
\]
**Step 2: Observe the Properties of the Coefficients**
Notice that the coefficie... | 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 givenennonary expression, I recognize that it resembles the expansion of a binomial expansion. The expression can be rewritten as the sum of two binomials that are now considered the same as the sum of the products of three real numbers \\( xy + yz + zx \\).",
"The binomial expansion of \\( \\binom{... | To find the value of the expression
$$
\frac{1}{1+x+xy} + \frac{1}{1+y+xy} + \frac{1}{1+z+zx},
\quad
and know that \( xy + yz + zx = 1 \), we can follow these steps:
1. **Express the Given Expression in a Binomial Expansion:**
Start by recognizing the structure of the given expression. Both terms will be combined ... | 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 values of \\( b \\) that make both functions \\( f(x) = x^3 - 3x + b \\) and \\( g(x) = x^2 + bx - 3 \\) have a common root where their coefficients are equal.",
"For the equation \\( f(x) = 0 \\), substituting \\( x = 0 \\) into \\( f(x) \\) gives:",
"\\[",
"0 = 0^3 - 3 \\cdot 0 + ... | To determine the sum of all possible values of \( b \) for which the conditions \( f(x) = 0 \) and \( g(x) = 0 \) lie for all real numbers \( x \), we analyze each function separately.
### Step 1: Analyzing \( f(x) = x^3 - 3x + b \)
For \( f(x) = 0 \):
\[
0 = x^3 - 3x + b
\]
Substitute \( x = 0 \):
\[
0 = (0)^3 - 3(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? | [
"Alright, so I've got this problem here about solving real numbers \\( b \\) where \\( g(x) \\) is a polynomial of degree 1. And I've got two equations that have actual roots: \\( f(x) = 0 \\) and \\( g(x) = 0 \\). That makes me think there must be a root at some point \\( x_0 \\) so I can solve equations for \\( b... | The sum of all possible values of \( b \) for which the equation \( f(x) = 0 \) has a common root is \( b + 0 = b \).
\[
\boxed{b + 0 = b}
\] | 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? | [
"Okay, so I have this problem where I need to find the sum of all possible values of b for which the equation f(x) = 0 and g(x) = 0 has a common root. Let me see what that means.",
"First, I know that if a function f(x) is equal to zero, its graph is a parabola. Its graph is a straight line at y = 0 at its vertex... | The sum of all possible values of \( b \) for which the equations \( f(x) = 0 \) and \( g(x) = 0 \) have a common root is \(\boxed{0}\). | 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, I recognize that for two quadratic functions \\( f(x) = x^3 - 3x + b \\) and \\( g(x) = x^2 + bx - 3 \\) to have a common root, their sum must be zero. This is because the roots of a quadratic equation \\( x^2 + bx + c = 0 \\) must be equal.",
"Setting the sum of the functions equal to zero gives:",
"\\... | To determine the sum of all possible real values of \( b \) for which the equations \( f(x) = 0 \) and \( g(x) = 0 \) have a common root at the simultaneous point \( x = 4 \) and \( x = -1 \), follow these steps:
1. **Set the Sum of Functions to Zero:**
For a quadratic function \( f(x) = ax^2 + bx + c = 0 \) to... | 0 | true |
Open-Reasoner-Zero/orz_math_57k_collection | There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$ . Find $(m_1 + n_1) + (m_2 + n_2)... | [
"First, I need to simplify the given equation \\(64\\sin^2(2x) + \\tan^2(x) + \\cot^2(x) = 46\\). Recognizing that the cosine of \\(2x\\) and \\(\\cot^2(x)\\) are both positive, I can identify the expression as simply \\(46\\sin^2(2x)\\).",
"Next, I'll compare this simplified equation to the standard form of a si... | To solve the problem, let's simplify and analyze the given equation step by step.
**Given Equation:**
\[
64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46
\]
**Step 1: Simplify the Surrounding Sine**
Notice that \(\tan^2(x) = 2x\). This allows us to rewrite the equation as:
\[
64\sin^2(2x) + \tan^2(x) = 46 \cdot 2x = 92x
\]... | 100 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $a_1=-1$ , $a_2=2$ , and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$ . What is $a_{2006}$ ? $
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | [
"First, identify the given values: \\( a_1 = -1 \\), \\( a_2 = 2 \\), and \\( a_n = \\frac{a_{n-1}}{a_{n-2}} \\) for \\( n \\geq 3 \\).",
"Calculate \\( a_{n-1} \\) using the recursive formula:",
"\\[",
"a_{n-1} = \\frac{a_{n-2}}{a_{n-1}} = \\frac{2}{-1} = -2",
"\\]",
"Next, determine \\( a_{2006} \\) by ... | To find \( a_{2006} \) using the recurrence formula:
\[
a_n = \frac{a_{n-1}}{a_{n-2}} \quad \text{for } n \geq 3
\]
given \( a_1 = -1 \), \( a_2 = 2 \), and \( n \geq 3 \):
1. **Calculate \( a_{n-1} \):**
\[
a_{n-1} = \frac{a_{n-2}}{a_{n-1}} = \frac{2}{-1} = -2
\]
2. **Find \( a_{2006} \):**
\[
a_{2006} = \frac{a... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | $a_1=-1$ , $a_2=2$ , and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$ . What is $a_{2006}$ ? $
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | [
"To find \\( a_{2006} \\), I start by understanding the recursive pattern provided. The first term \\( a_1 = -1 \\) and the second term is \\( a_2 = 2 \\). The recursive formula is \\( a_n = \\frac{a_{n-2}}{a_{n-1}} \\).",
"I need to determine the 2006 terms up to \\( a_{2006} \\). Using the formula, each term is... | To determine \( a_{2006} \), we'll use the given recursive pattern:
\[
a_n = \frac{a_{n-1}}{a_{n-2}}
\]
**Step 1:** Start with the given initial values.
\[
a_1 = -1, \quad a_2 = 2
\]
**Step 2:** Determine the number of terms up to \( a_{2006} \).
\[
n_{2006} = 2005
\]
**Step 3:** Calculate the 2000th term \( a_{200... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the largest possible value in the real numbers of the term $$ \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} $$ with $x^2 + y^2 \ne 0$ . | [
"First, I recognize that the given expression \\(\\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}\\) can be simplified by factoring out common terms.",
"I notice that there are numbers that can be factored as perfect squares. Specifically, \\(3 = 3 \\times 1\\) and \\(16 = 2^4\\), which can be factored into \\((3 \\times 2... | To find the largest possible value in the real numbers of the term
\[
\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2},
\]
follow these steps:
1. **Factor the Left-Hand Side:**
Notice that the expression can be factored as a product of two perfect squares:
\[
\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} = 3(x^2 + y^2) + 1... | 19 | 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 determine the quintile of \\( f(x) \\) that satisfies \\( f(p) = 418{,}195{,}493 \\), we start by relaxing the constraints of the problem. The expression \\( p \\) can be any integer, including zero.",
"Since \\( f(x) \\) is a quartic function, it is corect to its form from the second-to-last row of the stand... | To find the quintile of \( f(x) \) that satisfies \( f(p) = 418{,}195{,}493 \), we first relax the constraints of the problem. Let the quintile be \( y \). We need to find all integers \( p \) such that \( f(p) = 418{,}195{,}493 \). Since \( f(x) \) is a quartic function, it is corect to its form from the second-to-las... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$ . There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before enteri... | [
"First, identify the corners of the rectangular pool table where the ball will hit.",
"The table has vertices at (0, 0) (12, 0), (12, 10) (0, 10), and (0, 10) (12, 10).",
"Path Analysis:",
"1. The center of the pool table is (0, 0). The first corner where the ball is hit is at (0, 0).",
"2. The spine of the... | To determine the number of walls that the ball bounces off before entering a pocket, let's analyze the pool table step by step.
### **Step 1: Identify the Corner Points**
We have the table as follows:
- **Vertex A**: $(0, 0)$
- **Vertex B**: $(12, 0)$
- **Vertex C**: $(0, 10)$
- **Vertex D**: $(12, 10)$
- **Vertex E*... | 9 | 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 number of black unit squares in the 9x9 square. Since each square has an area of 9 units, the total surface area is \\(9^2 = 81\\) square units. To find the number of black squares, I divide the total area by the area of one black square: \\(\\frac{81}{9} = 9\\).",
"Next, I'll cons... | To determine the number of black unit squares in the \(9 \times 9\) square, follow these steps:
1. **Calculate the Total Surface Area of the Square:**
The square has an area of \(9 \times 9 = 81\) square units.
2. **Determine the Number of Black Square:**
Each black square has an area of \(\frac{9}{9} =... | 27 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$ . Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$ , where $Q(x) = x^2 + 1$ . | [
"First, I recognize that the polynomial \\( P(x) = x^5 - x^2 + 1 \\) has 5 real roots \\( r_1, r_2, r_3, r_4, r_5 \\).",
"To find the product \\( Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5) \\), I can use the relationship between the roots and the coefficients of the polynomial. Specifically, if \\( P(r_1)Q(r_1)P(r_1)Q(r_1)P(... | To find the value of the product \( Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5) \) where \( P(x) = x^5 - x^2 + 1 \) and \( Q(x) = x^2 + 1 \), follow these steps:
1. **Understand the Relationship Between Roots and Constant Terms:**
For a polynomial \( P(x) \) with roots \( r_1, r_2, r_3, r_4, r_5 \), the product of the roots \(... | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | [
"To determine the smallest number of integers \\( k \\) such that \\( P(k) \\) has nonnegative coefficients, we start by recalling the minimal polynomial of \\( P(k) \\). The minimal polynomial is the second root on \\( \\mathbb{Q}^n \\), where \\( n \\) is the number of terms in the polynomial. For \\( k = 4 \\), ... | To determine the smallest number of integers \( k \) such that \( P(k) \) has nonnegative coefficients, we follow a systematic approach:
1. **Understanding Polynomial Minimalism:**
- The minimal polynomial of \( P(k) \) is often called its "first root" or interesting lower degree (\( d \)). For \( k = 4 \), the fir... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | [
"To determine the smallest number of steps Danica needs to find the polynomial Tatjana imagined, I recognize that she starts with an initial value for the polynomial, \\( P(x) \\), and then gradually increases \\( k \\) by each step.",
"Since she increases \\( x \\) by 1, her new value is \\( P(k + 1) \\). To fin... | To determine the smallest number of steps Danica needs to find the polynomial \( T(x) \) generated by the initial coefficients, let's analyze the given information step by step.
1. **Initial Polynomial:**
Let the initial polynomial be \( P(x) \) with nonnegative integer coefficients.
2. **Pavdanov Effect:**
... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | [
"To determine the smallest number of integers \\( k \\) such that Tatjana's polynomial is nonnegative, we need to find the range of \\( k \\) such that \\( P(k) \\geq 0 \\).",
"Given the equation \\( P(x) = x^2 \\), the polynomial is errorless and contains no nonnegative coefficients. The smallest nonnegative int... | To determine the smallest number of integers \( k \) such that the polynomial \( P(k) \) has nonnegative coefficients, we need to find the range of \( k \) such that \( P(k) \geq 0 \).
### Step-by-Step Solution:
1. **Understand the Problem:**
- Tatjana is using a polynomial \( P(x) \) with nonnegative integer coef... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers, $a$ and $b$ , are eliminated in $S$ , and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$ .
After doing this operation $99$ times, there's only $1$ number on $S$ . What val... | [
"First, I recognize that the set \\( S \\) initially has 100 numbers. When removing two numbers, \\( a \\) and \\( b \\), the remaining numbers are \\( 100 - a + b \\).",
"Next, I consider the use of multiplication and summation. When we change the order of \\( a \\) and \\( b \\), the operations can be reversed,... | To determine the possible values for the number \( a + b + ab \) that immediately make the set \( S \times [100] \) complete, let's analyze the problem step by step.
### **Initial Setup**
- **Set \( S \):** 100 numbers, following the sequence of numbers to be eliminated2.
- **Setting Up the Operations:**
- Remove \(... | 100 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers, $a$ and $b$ , are eliminated in $S$ , and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$ .
After doing this operation $99$ times, there's only $1$ number on $S$ . What val... | [
"To determine the values that can take the number 100, we start with the initial arrangement. Since 99 of the 100 numbers are on S and there are 19 gaps between them, there are 19 combinations of 2 elements(a), 2 elements(b), but 99 gaps.",
"We start with and end with 1:",
"1. First gap: 11 elements. Arrange us... | To determine the values that can take the number \(100\), we start with and end with \(1\):
1. **First Gap:**
Use the equation:
\(1 + 1(a + b) = 100 + 1 = 101 + 1 = 100\)
However, the number corresponds to \(11\) elements. This group cannot be reduced.
2. **Second Gap:**
Use the equation:
\(1... | 100 | 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 sum \\( m + n \\), we start by examining the form of the exponents in each number.",
"For the number \\( 2^{1997} \\), the exponent allows us to write \\( 2 \\) as a power of 2. Specifically, \\( 2^{\\max(0,1997)} = 2^1997 \\). This means the exponent \\( m \\) is equal to 1997.",
"For the num... | To determine the sum \( m + n \), we analyze the form of the exponents in both numbers.
1. **For \( 2^{1997} \):**
The number \( 2 \) can be expressed as a power of 2:
\[
2 = 2^0 \quad \Rightarrow \quad 2^{\max(0, 1997)} = 2^1997
\]
Therefore, \( m = 1997 \).
2. **For \( 5^{1997} \):**
All p... | 1998 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
*Anonymous Proposal* | [
"Okay, let me try to solve this problem. So, the problem is asking for the sum of all prime numbers that can be written both as a sum of two prime numbers and a difference of two primes. Hmm, let's break this down.",
"First, I need to regard each prime number as a sum of two primes. Let me recall that the sum of ... | The sum of two consecutive primes between 2 and 3 (i.e., 2 and 3) is \(2 + 3 = 5\).
\boxed{5} | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares. | [
"To find the natural numbers \\( n \\ge 2 \\) such that their ring modulo \\( n \\) has exactly an element not equal to the sum of any two squares, I need to check each number from 1 to 3 modulo 9 to see if there exists an element that is not a sum of two squares.",
"Starting with 1:",
"- \\( 1 \\div 9 = 0 \\) ... | To find the natural numbers \( n \ge 2 \) such that their ring of integers modulo \( n \) has exactly an element that is not equal to the sum of any two squares, we need to examine each of the numbers \( 1, 2, 3, \) and \( 4 \) modulo \( n \) to see if there exists such an element.
Let's analyze each number:
1. **For... | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | [
"To solve the sum \\(\\sum_{k=0}^{\\infty} \\frac{2^{k}}{5^{2^{k}} + 1}\\), I recognize that it resembles the Binomial Expansion of \\((2 + 1)^4\\) because the upper term is \\(2^{2 + 4 + 1} = (2^{1} + 2^2) = 6 - 4 = 2\\). Therefore, I can rewrite the sum as \\(\\frac{2^{n+1}}{n}\\) where \\(n\\) is the number of t... | To determine the value of \(p + q\), let's analyze the given sum:
\[
\sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}} + 1}
\]
This expression resembles the **Binomial Expansion** of \((2 + 1)^4\). Specifically, it can be rewritten as the quadratic expression:
\[
\sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^k} + 1} = \frac{(2 + 1... | 5 | true |
Open-Reasoner-Zero/orz_math_57k_collection | In a year that has $365$ days, what is the maximum number of "Tuesday the $13$ th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days. | [
"To determine the maximum number of \"Tuesday the 13th\" in a year with 365 days, we need to consider the specific frequency of the week days:",
"- April has 30 days.",
"- June has 30 days.",
"- September has 31 days.",
"- November also has 31 days.",
"- February also has 28 days.",
"The weekdays are al... | To determine the maximum number of "Tuesday the 13th" in a year with 365 days, follow these steps:
1. **Understand the Number of Days per Month:**
- **April:** 30 days
- **June:** 30 days
- **September:** 31 days
- **November:** 31 days
2. **Calculate the Number of "Tuesday" Days":**
- Since all days a... | 3 | 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? | [
"Okay, let me try to figure out this problem. The question is asking for the minimum number of colours needed to color all positive real numbers such that no two adjacent numbers have different colours. Hmm, so first, let's parse each part.",
"The problem says \"whatever minimum number of colours is sufficient to... | The minimum number of colours needed is determined by analyzing the problem as follows:
- **Concerning positive real numbers**, each element must be a set of non-negative integers. Solutions are of color zero (i.e., zeros) and two elements.
- **Concerning negative real numbers**, certain pairs simultaneously pair the ... | 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 all positive integers \\( n \\) which satisfy the equation \\( n \\cdot 2^{n-1} + 1 \\) being a perfect square, we can follow these steps:",
"1. **Understand the Equation**: The given equation is a product of two binomials, so they are either perfect squares or also integers.",
"2. **Check Each Pa... | To determine all positive integers \( n \) such that the equation \( n \cdot 2^{n-1} + 1 \) is a perfect square, follow these steps:
1. **Understand the Equation**:
- The equation is a product of two binomials.
- We need to find all positive integers \( n \) that satisfy the equation.
2. **Check Each Pair of Nu... | 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. | [
"To determine all positive integers \\( n \\) such that \\( n \\cdot 2^{n-1} + 1 \\) is a perfect square, we can follow these steps:",
"1. **Understand the problem**: We need \\( n \\cdot 2^{n-1} + 1 \\) to be equal to a perfect square. The positive integer \\( n \\) must satisfy \\( 2^n - 1 < 1 \\).",
"2. **So... | To determine all positive integers \( n \) such that \( n \cdot 2^{n-1} + 1 \) is a perfect square, follow these steps:
1. **Understand the Problem:**
We need \( n \cdot 2^{n-1} + 1 \) to be a perfect square.
2. **Condition for a Perfect Square:**
A perfect square is a number whose square is an integer. The... | 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 $f(0), f(1), \dots, f(n-1)$ ? | [
"To determine how many integers \\( n \\leq 2020 \\) are relatively prime to the function \\( f(0), f(1), \\dots, f(n-1) \\), we need to find the prime factors of each number from 0 to 2020.",
"1. **Identify the prime factors for each number:**",
"- \\( 0 = 0 \\)",
"- \\( 1 \\)",
"- \\( 2, 3, 5, 7, 11, 13 \... | To determine how many integers \( n \leq 2020 \) are relatively prime to the function \( f(0), f(1), \dots, f(n-1) \), we need to find the prime factors of each number from 0 to 2020.
1. **Identify the prime factors for each number:**
- \( 0 = 0 \)
- \( 1 \)
- \( 2, 3, 5, 7, 11, 13 \)
- \( 6, 12, 18 \)
... | 11 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$ . Here we make the convention $\sqrt[1]{abc}=abc$ . | [
"First, I need to understand the inequality provided: \\( \\frac{a + b + c}{abc + 1} + \\sqrt[n]{abc} \\leq \\frac{5}{2} \\).",
"I know that \\( \\sqrt[1]{abc} = abc \\), which breaks the inequality for \\( a \\leq b \\leq c \\). This means that \\( a, b \\), and \\( c \\) must all be integers greater than 1 and ... | To find the largest positive integer \( n \) such that the inequality
\[
\frac{a + b + c}{abc + 1} + \sqrt[n]{abc} \leq \frac{5}{2},
\]
holds true for all \( a, b, c \in [0,1] \), follow these steps:
1. **Understand the Constraints:**
- The variable \( a, b, c \) must be integers greater than 1 and less than 5.
... | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$ . Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$ . If the lines $AD$ and $OE$ meet at $F$ , find $|AF|/|FD|$ . | [
"Okay, so I have this problem about a circle with center O, tangent to line AC at D, intersecting at E, and meeting at F. And I need to find the ratio of the distance from A to the point D to the distance from A to point E, which is denoted as |AF|/|FD|. Hmm, let me try to visualize this.",
"First, the circle has... | The ratio |AF|/|FD| is \(\boxed{\dfrac{1}{2}}\). | \frac{1}{2} | 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, let's see. The problem is about convex polygons and a square of side length 1. We need to show that the sum of the sides of P is less than or equal to 4. Hmm, convex polygons and squares. I remember something about triangle sums being bounded, so maybe P can't be contained within a triangle? Is that right? W... | Let \( P \) be a convex polygon \( P \) such that \( P \) lies in a square of side length 1. We need to show that the sum of the sides of \( P \) is less than or equal to 4.
1. **Geometry Property (Showing Triangle Sufficiency):**
- Any convex polygon \( P \) is internal if it lies on the perimeter of a square.
... | 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 convex polygon P cover up to one side of the square. Let me think about how to approach this step by step.",
"First, the problem is about a convex polygon P that's contained inside a square of side length 1. To get an idea, maybe I should think about the r... | The sum of the sides of a convex polygon that is contained within a square of side length 1 is always less than or equal to 4. This is because the polygon extends beyond the square completely, ensuring that all as-drawn sides occupy all matching square sides and do not overlap with an adjacent pair or one side of the s... | 4 | true |
Open-Reasoner-Zero/orz_math_57k_collection | What is the maximal number of solutions can the equation have $$ \max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$ where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$ . | [
"To determine the maximal number of solutions to the equation \\( a_1x + b_1 + a_2x + \\ldots + a_{10}x + b_{10} = 0 \\) for all real numbers \\( a_1, b_1, a_2, \\ldots, a_{10} \\) where each \\( a_i \\) is not equal to 0, follow these steps:",
"1. **Understand the Nature of the Equation**:",
"- The equation \\... | To determine the maximal number of solutions to the equation \( a_1x + b_1 + a_2x + \ldots + a_{10}x + b_{10} = 0 \) for all real numbers \( a_1, b_1, a_2, \ldots, a_{10} \) where each \( a_i \) is not equal to 0, follow these steps:
1. **Understand the Nature of the Equation**:
- The equation \( a_1x + b_1 + a_2x ... | 2 | true |
Open-Reasoner-Zero/orz_math_57k_collection | What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? | [
"To determine the largest number of integers in a set of positive integers between \\(1\\) and \\(100\\) that are divisible by another number, we can use the concept of Fibonacci numbers. A Fibonacci number is the sum of the first \\(n\\) natural numbers. Since \\(100\\) is an even number, we can divide it by \\(2\... | To determine the largest number of integers in a set of positive integers between \(1\) and \(100\) that are divisible by another number, we can utilize the concept of **Fibonacci numbers**. A Fibonacci number is the sum of the first \(n\) natural numbers.
Since \(100\) is an even number, we can divide it by \(2\) to... | 50 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$ , $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$ .
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(... | [
"To determine the smallest positive integer \\( N \\) such that Alice always knows for sure the parity of \\( f(0) \\) within set \\( \\{1,2, \\ldots, 187\\} \\), we follow these steps:",
"1. **Analyze the Parity of \\( f(0) \\):**",
"- By definition, \\( f(0) \\) must be an integer.",
"- Since \\( f \\) is a... | The smallest positive integer \( N \) so that Alice always knows for sure the parity of \( f(0) \) within set \( \{1,2, \ldots, 187\} \) is \( \boxed{187} \). | 187 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Evan has $10$ cards numbered $1$ through $10$ . He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$ , the remainder is $1$ . Find the maximum number of cards he could have chose.
*Proposed by Evan Chang* | [
"To determine the maximum number of cards Evan could have chosen, we can model the problem using combinatorics. Evan has cards numbered from 1 to 10. He takes the product of the numbers on his chosen cards and then divides the resulting product by 3. When this product is divided by 3, the remainder is 1. This impli... | To determine the maximum number of cards Evan could have chosen, let's analyze the situation step by step.
1. **Let \( n \) be the number of cards Evan chooses.**
2. **The product of the numbers on his chosen cards is \( n! \).**
3. **The product of the numbers on cards numbered 1 through 3 is \( 3! \).**
4. **When th... | 6 | 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 centers \\( A \\) and \\( D \\), which is \\( AC \\). Since the regular octagon has sides with 10 units each, solving the congruence \\( AD = AC \\) will allow me to find the lengths of all sides.",
"Next, I should consider how the circle intersects point \\( I \\)... | To determine the length of segment \( I \) in the regular octagon \( ABCDEF GH \) where:
- \( ABCDEFGH \) is a regular octagon, with each side measuring 10 units.
- Two circles intersect at point \( I \) with centers \( A \) and \( D \) and radius \( AC \).
- Points \( A \)$$Australia$$and \( D$$ \) are diametrically ... | 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, recognize that the octagon ABCDEFGH has 10 equal sides, which corresponds to 5 core segments. Each core segment has a radius of 10 units, so the total radius of the octagon is 50 units.",
"The circle passing through point I does not intersect the octagon except at points where it intersects the core segme... | To solve the problem, let's break it down step by step.
**Step 1: Understand the Octagon Structure**
A regular octagon with center \( A \) and radius \( AC \) is. This means:
- The set \( ABCDEFGH \) has **5 main core segments**, each spaced \( 10 \) units apart.
**Step 2: Identify the Core Segments**
From point \(... | 10 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$ .
Find $\sum_{n=0}^{1992} 2^nx_n$ . | [
"I need to find the sum of the roots of the quadratic equation \\( ax_n + bn = c \\) for the series given by \\( x_0 = 1992 \\), \\( nx_n = 2 \\), and \\( n \\geq 1 \\).",
"First, I recall that for a quadratic first derivative equation \\( ax^2 + bx + c = 0 \\), the sum of the roots is \\( -\\frac{b}{a} \\). In t... | To find the sum of the roots of the quadratic equation \( x_0 = 1992 \), we start by noting that for a quadratic first derivative equation \( ax^2 + bx + c = 0 \), the sum of the roots is \( -\frac{b}{a} \).
Given:
- \( x_0 = 1992 \)
- \( nx_n = 2 \)
- \( n \geq 1 \)
The sum of the roots is:
\[
\sum_{n=0}^{1992} 2^nx... | 1992 | 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 find the value of \\( a \\), \\( b \\), and \\( c \\) such that the equation \\( 2abc = a + b + c + 4 \\) holds true for all \\( a, b, c \\) nonzero integers.",
"Since \\( a, b, c \\) are nonzero, their sum \\( a + b + c \\) must be the only number that can be substituted into the equation witho... | To determine how many ordered triples \((a, b, c)\) satisfy the equation \(2abc = a + b + c + 4\), follow these steps:
1. **Understand the Problem:**
We need to find all triples \((a, b, c)\) such that:
\[
ab + bc + ca + 4 = 0
\]
For such triples, the sum of the elements must contain all nonzero numbers... | 6 | true |
Open-Reasoner-Zero/orz_math_57k_collection | For a positive integer $n$ , let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$ . Find the number of fair integers less than $100$ . | [
"To determine the number of fair integers \\( n \\) that are perfect squares and perfect cubes less than 100, let \\( s(n) \\) denote the number of perfect square divisors and \\( c(n) \\) denote the number of perfect cube divisors. A prime number \\( p \\) is considered a perfect square if the square root \\( p = ... | To determine the number of fair integers \( n \) that are perfect squares and perfect cubes less than 100, let \( s(n) = n^a/(n^b) \), where \( p(n) \) is the number of perfect square divisors and \( p(n) \) is the number of perfect cube divisors.
1. **Number of Perfect Square Divisors:**
\[
s(n) = (n + 1)(n a +... | 7 | true |
Open-Reasoner-Zero/orz_math_57k_collection | The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$ , $b$ , $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at t... | [
"To determine the largest possible number on the board that remains after nine moves, we can model the process using a lattice structure. Here, each move moves points to a new position on the board, subtracting the expected number of positions on the left.",
"1. **First Move**: Move 1 point to the 8th number in t... | To determine the largest possible number on the board that remains after nine moves, we can model the process using a lattice structure. Here, each move moves points to a new position on the board, subtracting the expected number of positions on the left.
1. **First Move**: Move 1 point to the 8th number in the sequen... | 8\sqrt{6} | true |
Open-Reasoner-Zero/orz_math_57k_collection | The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$ , $b$ , $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at t... | [
"Alright, so I've got this problem here about numbers from 1 to 10 on a whiteboard. I need to figure out the largest possible number left after every minute, where I erase every other number, and finally write the square root of the remaining numbers. Let me try to break it down step by step.",
"First, let's elim... | The major portion of the number remaining after erasing all other numbers is 9, 9, and 8. The final count is:
**9, 9, 8**
**Answer:** $\boxed{9, 9, 8}$ | 8\sqrt{6} | true |
Open-Reasoner-Zero/orz_math_57k_collection | Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\] | [
"Okay, so I need to prove that the sum 1/n of the unique reciprocals of the even numbers from 2 to 41 can have at least eight terms. Hmm, let me think.",
"First, the problem defines the arithmetic progression as all even numbers. Let's denote that second term as 2. Then the first term is 5, third term is 8, forth... | \boxed{\frac{1}{2} \cdot 1 \cdot 1 \cdot (4 \div (n^2)^n)^n = 8} | 8 | 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 lengths of sides \\( AD \\) and \\( BC \\) using the given Pythagorean theorem. Since \\( D \\) and \\( H \\) are midpoints of \\( BD \\), I can use the midpoint triangle triangle IDG to calculate \\( DH \\).",
"Using the midline theorem on \\( BD \\), I find that \\( DH = \\frac{B... | To find the ratio of the area of triangle \( EHG \) to that of quadrilateral \( ABCD \), follow these steps:
1. **Determine the lengths of sides \( AD \) and \( BC \) using the Pythagorean theorem:**
- Let \( D \) be the midpoint of \( BD \).
- Let \( H \) be the midpoint of \( BC \).
- Using the Pythagorean... | \frac{1}{4} | true |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \f... | [
"Okay, so I've got this problem here about a sphere touching to a regular tetrahedron and touching externally to a regular tetrahedron. I need to figure out the largest volume of a sphere that touches both the upper and bottom spheres. Hmm, let me try to break this down step by step.",
"First, what's a regular te... | The largest volume of the sphere that touches both the upper and bottom spheres is \( \frac{1}{3} \) cubic units.
\[
\boxed{\frac{1}{3}}
\] | \frac{1}{3} | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the minimum value of $$ \big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big| $$ for real numbers $x$ not multiple of $\frac{\pi}{2}$ . | [
"Okay, so I have this problem here: I need to find the minimum value of the expression sin(x + cos(x)) + tan(x) + sec(x) + cosec(x) for real numbers x not multiple of π/2. Hmm, complex numbers... Yeah, I remember from my notes that sin(x + cos) can be simplified using the tangent addition formula. Let me try to rec... | The minimum value of the expression \( \sin(x + \cos(x)) + \tan(x) + \sec(x) + \cosec(x) \) for real numbers \( x \) not multiple of \( \frac{\pi}{2} \) is found by recognizing that the expression simplifies to a continuous third-order function above \( 0 \). Here's the step-by-step derivation:
1. Recognize that \( \s... | 2\sqrt{2} - 1 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Find the minimum value of $$ \big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big| $$ for real numbers $x$ not multiple of $\frac{\pi}{2}$ . | [
"To find the minimum value of the expression \\(\\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\arcsin(x)\\) for real numbers \\(x\\), I start by recognizing that each trigonometric function can be rewritten as \\(\\setminus i\\).",
"This allows me to combine the trigonometric functions more effectively. By c... | To find the minimum value of the expression \(\sin x + \cos x + \tan x + \cot x + \sec x + \arcsin(x)\) for real numbers \(x\), we can use trigonometric methods. Here's a step-by-step solution:
### Step 1: Combine Precompositions
Each trigonometric identity can be combined by letting each function be a combination of ... | 2\sqrt{2} - 1 | 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. | [
"To find the sum of all possible sums \\( a + b \\) where \\( a \\) and \\( b \\) are nonnegative integers satisfying \\( 4^a + 2^b + 5 = (4+^a)^b + 5 \\) is a perfect square, we can express \\( b \\) in terms of \\( a \\):",
"\\[",
"b = \\log_4 {(4^a)} + \\log_4 {(2^b)}",
"\\]",
"Next, we add the logarithm... | To find the sum of all possible sums \( a + b \) where \( a \) and \( b \) are nonnegative integers satisfying \( 4^a + 2^b + 5 = (4+^a)^b + 5 \) is a perfect square, follow these steps:
1. **Express \( b \) in terms of \( a \):**
Since \( \log_4 {(4^a)} + \log_4 {(2^b)} = 4^a \) and also \( \log_4 {(2^b)} = \log_4... | 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 the value of f(9). The function f is defined such that whenever x plus y is an integer (which it is not in the range strictly greater than or equal to 8), then f(x + y) equals f(x * y). That means for any integer combination of x and y (which aren't more than ... | To determine the value of \( f(9) \), we use the defined function \( f \) where \( f(x + y) = f(xy) \). Since \( f(8) = 9 \) is given, \( f(x + y) \) must be equal to \( f(x \cdot y) \) for integer combination \( x + y \) not exceeding 4. This implies \( x \) and \( y \) are integers such that their sum is an integer, ... | 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 solve the problem, we start by analyzing the given function \\( f \\) and set conditions.",
"1. **Understanding the Function \\( f \\):**",
"- The function \\( f \\) is defined as \\( f(x\\) greater than or equal to 8 and \\( f(x+y)\\) greater than or equal to 9 \\). This implies that:",
"\\[",
"f(x + y... | To determine \( f(9) \) given the function \( f: X\rightarrow X \) with the constraints \( f(x\) \geq 8 \) and \( f(x + y) \geq 9 \), and knowing that \( f(8) = 9 \), follow these steps:
1. **Understand the Function \( f \):**
- The function \( f \) is defined as:
\[
f(x\) \geq 8 \quad \text{and} \quad f(... | 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) .$ | [
"Suppose we have the set \\( X \\) which contains all positive integers greater than or equal to 8. We are given that \\( f(x \\pm y) = f(xy) \\) for all \\( x \\geq 4 \\) and \\( y \\geq 4 \\).",
"We are also told that \\( f(8) = 9 \\).",
"Let's deduce that \\( f(-5) = -3 \\).",
"Now, since \\( f(x \\pm y) =... | Let's solve step by step the problem.
**Given:**
- \( X \) is the set of all positive integers greater than or equal to 8.
- \( f: X \rightarrow X \) is a function for all \( x \geq 4 \) and \( y \geq 4 \).
- \( f(x \pm y) = f(xy) \) for all \( x \geq 4 \) and \( y \geq 4 \).
- \( f(8) = 9 \).
**Our Goal:**
Find \( f... | 9 | 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, let's tackle problem two. Hmm, problem two involves a function with real-valued constant applications, right? The problem states that f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), and the question is asking for a positive integer for which the sum of f(a) and f(1995) is equal to f(a+1) + f(a+2) for all... | The equation \( a+1 + a+2 = f(a+1)(a+2) \) holds for all positive integers \( a \). Substituting \( a = 3 \) gives a=3:
\boxed{3} | 3 | true |
Open-Reasoner-Zero/orz_math_57k_collection | Square $S_1$ is inscribed inside circle $C_1$ , which is inscribed inside square $S_2$ , which is inscribed inside circle $C_2$ , which is inscribed inside square $S_3$ , which is inscribed inside circle $C_3$ , which is inscribed inside square $S_4$ .
[center]<see attached>[/center]
Let $a$ be the side len... | [
"Okay, so I have this problem here about Square S1 inscribed in Circle C1, and Square S2 inscribed in Circle C1, etc. I need to find a fraction a/b from the given side lengths a and b, where a and b are the sides of Square S1. Hmm, let me think about how to approach this.",
"First, I remember that in a scenario w... | Given a circle of radius \( b \) inscribed in side \( a \) of Square S1, and another circle of radius \( b \) inscribed in side \( a \) of Square S2, and another circle of radius \( b \) inscribed in side \( a \) of Square S3, we need to find the ratio \( \frac{a}{b} \).
Since all the sides of Square S1 are parallel t... | 2\sqrt{2} | true |
Open-Reasoner-Zero/orz_math_57k_collection | Square $S_1$ is inscribed inside circle $C_1$ , which is inscribed inside square $S_2$ , which is inscribed inside circle $C_2$ , which is inscribed inside square $S_3$ , which is inscribed inside circle $C_3$ , which is inscribed inside square $S_4$ .
[center]<see attached>[/center]
Let $a$ be the side len... | [
"Alright, so I have this problem aboutsquares inscribed in circles, and I need to figure out the ratio between the side lengths ofsquares 1, 2, and 3. Let me try to visualize this and break it down step by step.",
"First, let's assign variables to the side lengths. Let's say a is the side length of square 1, and ... | The ratio of the side lengths of squares 1, 2, and 3 is **2:1**.
**Step-by-Step Explanation:**
1. **Assign Variables:**
- Let \( a \) be the side length of square 1.
- Let \( b \) be the side length of square 2.
2. **Analyze Square 3:**
- Square 3 is inscribed in circle \( C3 \), so its side length \( a \) ... | 2\sqrt{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... | [
"Alright, let's see. The problem is about a four-digit number that's *literally 1434*, which is right where digits 1, 4, 3, and 4 come together as patterns because when you shift them around, the result is 1434. So, possibly 1434 is actually 1430, 1431, 1432, 1433, 1434, etc. The possible numbers are 1, 6, 4, 3, 8,... | The possible four-digit numbers that have exactly prefix 1 for their digits and when modulated by the remainder when divided by 5, result in 1434 are:
\[1, 6, 4, 3, 8, 4, 9.\]
The sum of these numbers is:
\[16. \]
\boxed{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... | [
"Okay, so I have this problem here about a four-digit number, n, and it's a *literally 1434* if you replace any digit with the remainder when 9 is divided by 5. So, the result is 1434. Let me translate that into math terms to make it clearer.",
"If each digit of n is replaced by a number that's exactly divisible ... | The four-digit numbers that are literally 1434* are 1434, 1439, 1438, and 1437, totaling 67384.
**Answer:** \boxed{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, I need to identify the four-digit numbers that are literally 1434 by mapping the respective remainders when each division by 5 is done.",
"1. **Interestingly:** In integer multiplication, divisibility by 10 is not a requirement for a four-digit number, as the corresponding remainders for each half of the ... | The four-digit numbers that are literally 1434 by using remainders when divided by 5 are:
1. $\overline{abcd}$
2. $\overline{cddb}$
3. $\overline{dcde}$
4. $\overline{fghji}
5. $\overline{gghi}$
6. $\overline{gjhix}
7. $\overline{ihjf}
Each of these numbers is uniquely determined by the fact that the four-digit numbe... | 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 $1/2$ | [
"Alright, let's try to work through this problem. So, the problem is about the sides of a triangle with area 1/2, given that there are ten Fibonacci numbers within the Fibonacci sequence. The Fibonacci numbers are defined as F_n = F_{(n+1)} + F_{n}, where n starts at 1.",
"First, I remember that the areas of tria... | The sides \( V_n, V_{n+1}, V_{n+2} \) are indeed an open triangle with height \(\frac{1}{1}\), area \(\frac{1}{2}\).
\boxed{\text{Area }= \frac{1}{2}} | \frac{1}{2} | 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 prove that the triangle area of the Fibonacci sequence with order n is equal to the squares of its Fibonacci numbers. Hmm, let me start by understanding what Fibonacci numbers are used for. The Fibonacci sequence starts with F_1 and F_2, then F_3 and F_4, and so on. I remember that some of these... | \boxed{V_n = V_n + F_n \text{ and } V_{n+1} = F_n + F_n \text{ each resulting in the triangle for \( F_1, F_2, F_3, F_4, F_5 \) is the sides of a triangle with area } \boxed{\frac{1}{2}} \). | \frac{1}{2} | true |
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