messages listlengths 4 20 | tools listlengths 1 1 |
|---|---|
[
{
"content": "There are 152 students at Dala High School. Assume the following: \n- 100 students take a Math class \n- 94 students take a Science class \n- 57 students take an English class \n- 73 students take a Math class and a Science class \n- 24 students take a Math class and an English class \n- 27 ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find all triples of three-digit positive integers $x < y < z$ with $x, y, z$ in arithmetic progression and $x, y, z + 1000$ in geometric progression. The original answer is in the form of (k, m, n), please give the value of k + m + n.\n\n[i]For this problem, you may use calculators or computers to... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let\n$a=\\frac{1^2}{1} + \\frac{2^2}{3} + \\frac{3^2}{5} + \\; \\dots \\; + \\frac{1001^2}{2001}$\nand\n$b=\\frac{1^2}{3} + \\frac{2^2}{5} + \\frac{3^2}{7} + \\; \\dots \\; + \\frac{1001^2}{2003}.$\nFind the integer closest to $a-b$.\n\nRemember to place the final answer in the last part using the... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the greatest integer less than $\\sqrt{10} + \\sqrt{80}$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nI need to find the greatest integer le... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The legs of a right triangle measure 6 inches and 8 inches. Find the greatest area of a rectangle that can be inscribed in the right triangle( square inches). Give the exact answer\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "ABCDE is a pyramid with square base ABCD. Point E is directly above A with AE = 1024 and AB = 640. The pyramid is cut into two pieces by a horizontal plane parallel to ABCD. This horizontal plane is a distance h above the base ABCD. The portion of ABCDE that is above the plane is a new pyramid. Fo... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many pairs of positive integers $(a,b)$ with $a + b \\le 100$ satisfy the equation \\( \\frac{a + \\frac{1}{b}}{\\frac{1}{a} + b} = 13 \\)?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_call... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "When the roots of the polynomial \n\n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\n\nare removed from the number line, what remains is the union of $11$ disjoint open intervals. Find the number of these intervals on which \\( P(x) \\) is positive.\n\nRemember to place the... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $a + \\frac{1}{b} = 8$ and $b + \\frac{1}{a} = 3$. Find the sum of the two possible real values for $a$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In the figure below, choose point \\(D\\) on \\(\\overline{BC}\\) so that \\(\\triangle ACD\\) and \\(\\triangle ABD\\) have equal perimeters. What is the area of \\(\\triangle ABD\\)?\n\n[asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label(\"$A$\", (0,0), SW); label(\"$B$\", (4,0), ESE); label(\"$C$\", (... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "On the trip home from the meeting where this $AMC~10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $bbcac,$ where $0 \\le a < b < c \\le 9,$ and $b$ was the average of $a$ and $c.$ How many different five-digit numbers satisfy all these propertie... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form as [a, b). Please give the value of 2a + 2b.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'T... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A finite sequence of positive integers $ m_i$ for $ i=1,2,...,2006$ are defined so that $ m_1=1$ and $ m_i=10m_{i-1} +1$ for $ i>1$. Let $ k$ be the number of these integers that are divisible by $ 37$. Find the value of $ k$.\n\nRemember to place the final answer in the last part using the format... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In a triangle $ABC$, point $D$ lies on $AB$. It is given that $AD = 25$, $BD = 24$, $BC = 28$, and $CD = 20$. Find the length of $AC$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many three-digit numbers leave a remainder of $2$ when divided by $5$ and a remainder of $7$ when divided by $9$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"con... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an integer.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The roots of the polynomial \\[x^3 - 52x^2 + 581x - k\\]are distinct prime numbers. Find $k.$\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nLet me ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content":... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Two circles have radii of 5 and 26, respectively. The smaller circle passes through the center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?\n\nRemember to place the final answer in the ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the value of \\(i + 2i^2 + 3i^3 + \\ldots + 2002i^{2002}\\). The original answer is in the form of \\(k + mi\\), please give the value of \\(k + m\\).\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$\n\\[\\sqrt{\\frac{a}{b + c + d}} + \\sqrt{\\frac{b}{a + c + d}} + \\sqrt{\\frac{c}{a + b + d}} + \\sqrt{\\frac{d}{a + b + c}} > m.\\]\n\nRemember to place the final answer in the last part using the format... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $S$ be the set of 10-tuples $(a_0, a_1, \\dots, a_9),$ where each entry is 0 or 1, so $S$ contains $2^{10}$ 10-tuples. For each 10-tuple $s = (a_0, a_1, \\dots, a_9)$ in $S,$ let $p_s(x)$ be the polynomial of degree at most 9 such that\n\\[p_s(n) = a_n\\]for $0 \\le n \\le 9.$ For example, $... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find all functions f from the set of real numbers into the set of real numbers which satisfy for all real $x,y$ the identity\n$f(xf(x+y))=f(yf(x))+x^2.$\nThe original answer is in the form of af(x) and bf(x), please give the value of a + b.\n\nRemember to place the final answer in the last part us... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "99 dwarfs stand in a circle. Some of them wear hats. There are no adjacent dwarfs wearing hats, and no two dwarfs wearing hats have exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs wearing hats?\n\nRemember to place the final answer in the last part using the ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many pairs of positive integers $(a,b)$ are there such that ${gcd}(a,b)=1$ and $\\frac{a}{b} + \\frac{14b}{9a}$ is an integer?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $8$ and $10$ through $20$ and has one blank face. When the two dice are rolled, what is the probability that the sum ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the path passes through the center of the plot. Calculate the number of square feet in the remaining grass area. The original answer is in the form $k\\pi - m\\sqrt{n}$, pleas... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $\\varphi$ be the positive solution to the equation $x^2 = x + 1$. For $n \\ge 0$, let $a_n$ be the unique integer such that $\\varphi^n - a_n \\varphi$ is also an integer. Compute $\\sum_{n=0}^{10} a_n$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'Th... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"co... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Given that $a_{i} \\in \\{0,1,2,3,4\\}$ for every $0 \\le i \\le 9$, if $6\\sum_{i=0}^{9}a_{i} 5^{i} \\equiv 1 \\pmod{5^{10}}$, find the value of $a_{9}$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $x^2+bx+c = 0$ be a quadratic whose roots are each two more than the roots of $3x^2-5x-7$. What is $c$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<t... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the smallest positive integer $n$ for which $9n-2$ and $7n + 3$ share a common factor greater than $1$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content":... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "If $a, b, c \\ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$, then what is the value of $a+b+c$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "For each integer $n \\ge 1$ define $a_n = \\left[\\frac{n}{\\left[\\sqrt{n}\\right]}\\right]$ (where $[x]$ denoted the largest integer not exceeding $x$, for any real number $x$). Find the number of all $n$ in the set $\\{1, 2, 3, \\cdots , 2010\\}$ for which $a_n > a_{n+1}$\n\nRemember to place t... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\\frac{m}{n}$ , ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In the coordinate plane, let $F = (5,0).$ Let $P$ be a point, and let $Q$ be the projection of the point $P$ onto the line $x = \\frac{16}{5}.$ The point $P$ traces a curve in the plane, so that\n\\[\\frac{PF}{PQ} = \\frac{5}{4}\\] for all points $P$ on the curve. Find the equation of this curve i... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Suppose $a_{1} < a_{2}< \\cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \\cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two se... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A small bottle of shampoo can hold $35~ milliliters$ of shampoo, whereas a large bottle can hold $500~ milliliters$ of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?\n\nRemember to place the final answer... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A triangle has sides that are 3, 4 and 5 units long. What is the radius of the inscribed circle?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nI ne... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many solutions of the equation $\\tan x = \\tan \\tan x$ are on the interval $0 \\le x \\le \\tan^{-1} 942$? (Here $\\tan^{-1}$ means the inverse tangent function, sometimes written $\\arctan$.)\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final a... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $f(x)=x+4$ and $g(x)=\\sqrt[3]{x}$. Find $(g^{-1} \\circ f^{-1})(6)$\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "I need to solve $(g^{-1} \\circ f^{-1... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the smallest 3-digit number of the form $aaa$ such that it is divisible by 9? (Here, $a$ represents a digit.)\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"con... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$\n\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final a... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \n$\\begin{aligned}&\\mathrm{Row 0:}&&&&&&&&&&1\\\\&\\mathrm{Row 1:}&&&&&&&&&1&&1\\\\&\\mathrm{Row 2:}&&&&&&&&1&&2&&1\\\\&\\mathrm{Row 3:}&&&&&&&1&&3&&3&&1\\\\&\\mathrm{Row... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let \\(T\\) be the set of ordered triples \\((x,y,z)\\) of nonnegative real numbers that lie in the plane \\(x+y+z=1\\). Let us say that \\((x,y,z)\\) supports \\((a,b,c)\\) when exactly two of the following are true: \\(x\\ge a, y\\ge b, z\\ge c.\\) Let \\(S\\) consist of those triples in \\(T\\)... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the value of the expression $1+3+5+\\cdots+2017+2019-2-4-6-\\cdots-2016-2018$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nI need to find... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $p(x)$ be the polynomial with leading coefficent 1 and rational coefficents, such that \\[p\\left(\\sqrt{3 + \\sqrt{3 + \\sqrt{3 + \\ldots}}}\\right) = 0,\\] and with the least degree among all such polynomials. Find $p(5)$.\n\nRemember to place the final answer in the last part using the for... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The sequence $a_1, a_2, \\cdots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\\log_8 a_1+\\log_8 a_2+\\cdots+\\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$\n\nRemember to place the final answer in the last part us... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The curve $y = \\sin x$ cuts the line whose equation is $y = \\sin 70^\\circ$ into segments having the successive ratios\n\\[\\dots p : q : p : q \\dots\\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p, q)$. The original answer is in the format $(p, q)$, please gi... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A point $X$ exactly $\\sqrt{2} - \\frac{\\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\\triangle XYZ$ is an equilateral triangle can be... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins equal to the difference between the number of stones in the box the stone was put in and the box the stone was taken from (the moved stone does not cou... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the number of 8-tuples of binary inputs $\\{A. B.C,DE.F,G,H\\}$ such that \n$$\\begin{aligned} & \\{(A AND B) OR (C AND D)\\} { AND }\\{(E AND F) OR (G AND H)\\} \\\\ & =\\{(A OR B) AND (C OR D)\\} OR \\{(E OR F) AND (G OR H)\\} \\end{aligned}$$ \nThe AND gates produc... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the least positive integer $n$ such that $\\cos\\frac{\\pi}{n}$ cannot be written in the form $p+\\sqrt{q}+\\sqrt[3]{r}$ with $p,q,r\\in\\mathbb{Q}$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The points A(5,-8), B(9, -30) and C(n,n) are collinear (that is, lie on the same straight line). What is the value of n?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In four years Kay will be twice as old as Gordon. Four years after that Shaun will be twice as old as Kay. Four years after that Shaun will be three times as old as Gordon. How many years old is Shaun now?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The f... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many ways are there to represent 100000000 as the product of three factors? Factorizations which differ in the order of the factors are considered to be distinct.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role":... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \n\\begin{align*} x^2+3y&=9 \\\\ (|x|+|y|-4)^2 &= 1 \\end{align*}\nFind the number of solutions.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes he... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "How many square units are in the region satisfying the inequalities $y \\ge |x|$ and $y \\le -|x|+3$? Express your answer as a decimal, then multiply the result by 10 and provide the final value.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answe... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "I need to solve for $a^4 + b^4$ where $a... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\\overline{AB}$ and two lines parallel to $\\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the centr... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point? The original answer is in the format $(0, y)$. Please provide the value of $y$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "For every integer $n ≥1$ let $t_{n}$ denote the n-th triangular number, defined by $t_{n}=\\frac{n(n+1)}{2}$. Find the values of $n$ for which $$\\frac{1^2+2^2+\\ldots+n^2}{t_1+t_2+\\ldots+t_n}$$ is an integer number.\n\nRemember to place the final answer in the last part using the format: \n\n\\b... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is, each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "At most how many circles of radius 1 cm can fit inside a square with side 20 cm such that the circles do not overlap?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"con... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A circle has a circumference of $8\\pi$. Determine its radius.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\n\nI need to find the radius of a circl... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the product of two consecutive factorials from the numbers $98!$, $99!$, $100!$, and $101!$ that is a perfect square. The original answer is in the form of $k! \\cdot (k+1)!$. Please give the value of $k$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "There exists a polynomial \\(P\\) of degree 5 with the following property: If \\(z\\) is a complex number such that \\(z^5 + 2004z = 1,\\) then \\(P(z^2) = 0.\\) Calculate\n\\[\\frac{P(1)}{P(-1)}.\\]\nThe original answer is in the form of \\(-\\frac{k}{m}\\). Please give the value of \\(k + m\\).\... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Meena owns a bottle cap collection. During her vacation, her collection size increases by $40\\%$. Later, she gives away $20\\%$ of her current collection. After the vacation, Meena has $21$ more bottle caps than she had before. How many bottle caps did Meena have before her vacation?\n\nRemember ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the value of x satisfying the equation $\\frac{2}{1300 x-2600}=\\frac{1}{650}-\\frac{1}{651}$?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "I'll so... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the largest integer $k$ such that $12^k$ divides $66!$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "I'll solve this problem step by step, finding the... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $a, b, c$ be three different numbers such that $$\\left\\{ \\begin{array} { l l } { \\boldsymbol { a } ^ { 3 } + \\mathbf { 7 } \\boldsymbol { a } + \\mathbf { 1 1 } = \\mathbf { 0 }, } \\\\ { \\boldsymbol { b } ^ { 3 } + \\mathbf { 7 } \\boldsymbol { b } + \\mathbf { 1 1 } = \\mathbf { 0 }, }... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $f(x)=x^{6}-3 x^{4}+5 x^{2}-7$. If $f(7)=110684$, compute f(-7)\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nLet me analyze this problem step ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\\overline{AE},$ and through $E,$ draw a line $... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Ant Amelia starts on the number line at \\(0\\) and crawls in the following manner. For \\(n=1,2,3,\\) Amelia chooses a time duration \\(t_n\\) and an increment \\(x_n\\) independently and uniformly at random from the interval \\((0,1).\\) During the \\(n\\)th step of the process, Amelia moves \\(... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\\frac{1}{3}$ of them and then gave $\\frac{1}{4}$ of what remained to her friend Rachel. Tracy and her mom then each ate 15 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find all prime numbers $p$ such that $2^p + p^2$ is also a prime number.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nI need to find all prime num... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Calculate the sum of the expression $$10^0 + 10^1 + 10^2 + 10^3 + 10^4 + 10^5$$ and find the power of 10 that is closest to this sum. Write your answer as an integer representing the exponent of the power of 10.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the maximum value of $e^{\\sin x \\cos x \\tan x}$. If the answer is in the form $k \\cdot e$, please provide the value of $k$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the number of triples of nonnegative integers $(x, y, z)$ such that \n\\[x^2 + 2xy + y^2 - z^2 = 9.\\]\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<t... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "For every positive integer $n$, the function $f$ is defined as follows:\n- $f(2n+1) = 2f(2n)$\n- $f(2n) = f(2n-1) + 1$\n- $f(1) = 0$\n\nDetermine the remainder when $f(2005)$ is divided by $5$.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "For some positive integers $k$ and $p,$\n\\[\\sqrt{23 + \\sqrt{28}} + \\sqrt{23 - \\sqrt{28}} = \\sqrt{k + \\sqrt{p}}.\\]\nThe original answer is in the format (k, p). Please provide the value of k + p.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The fina... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In the expression $c \\cdot a^b - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes her... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "If $\\frac{\\sin (2A + B)}{\\sin B} = 5,$ then find $\\frac{\\tan (A + B)}{\\tan A}.$The answer is in the form \\frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the sum of all real values of x such that $$1+x+x^{2}+\\cdots+x^{2020}=2021{\\sqrt{x}}^{2020}$$\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "I'll solv... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter \\(10\\) and altitude \\(12\\), and the axes of the cylinder and cone coincide. Find the radius of the cylinder. The original answer is in the form \\(\\frac{k}{m}\\), where... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In triangle $ABC,$ $\\sin A = \\frac{3}{5}$ and $\\sin B = \\frac{24}{25}.$ The original answer consists of two possible values for $\\sin C$ in the form $k/m$ and $n/p$, where each fraction is in its simplest form. Please provide the sum of $k, m, n,$ and $p$.\n\nRemember to place the final answe... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let \\(A_1A_2 \\ldots A_{12}\\) be a regular dodecagon. Equilateral triangles \\(\\vartriangle A_1A_2B_1\\), \\(\\vartriangle A_2A_3B_2\\), \\(\\ldots\\), and \\(\\vartriangle A_{12}A_1B_{12}\\) are drawn such that points \\(B_1\\), \\(B_2\\), \\(\\ldots\\), and \\(B_{12}\\) lie outside dodecagon ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'T... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation \n$8(\\log_n x)(\\log_m x)-7\\log_n x-6 \\log_m x-2013 = 0$ \nis the smallest possible integer. What is the value of $m+n$?\n\nRemember to place the final answer in the last part using the format: \... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "The whole numbers from 1 to $2k$ are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of $k$?\n\nRemember to place the final answer in the last part using the format: \n\n\\box... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$:\n\\[\n\\begin{align*}\n6^... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.\n\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"cont... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Find the smallest positive integer $n$ such that if $n = p_1^{a_1} p_2^{a_2} \\ldots p_s^{a_s}$, then \\[ \\phi(n) = n \\left(1 - \\frac{1}{p_1}\\right) \\left(1 - \\frac{1}{p_2}\\right) \\ldots \\left(1 - \\frac{1}{p_s}\\right) = \\frac{2^5}{47}n. \\]\n\nRemember to place the final answer in the ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "In rectangle \\(ABCD\\), angle \\(C\\) is trisected by \\(\\overline{CF}\\) and \\(\\overline{CE}\\), where \\(E\\) is on \\(\\overline{AB}\\), \\(F\\) is on \\(\\overline{AD}\\), \\(BE=6\\), and \\(AF=2\\). Find the area of \\(ABCD\\). The original answer is in the form \\(k\\sqrt{3} - m\\). Plea... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the area of the region in the coordinate plane defined by the inequality \\[\\left||x|-1\\right|+\\left||y|-1\\right|\\leq 1?\\] Find the area.\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"t... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Four pens and three pencils cost $\\$2.24$. Two pens and five pencils cost $\\$1.54$. No prices include tax. In cents, what is the cost of a pencil?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "Let \\(a, b, c, d\\) be rational numbers with \\(a > 0\\). If for every integer \\(n \\ge 0\\), the number \\(an^{3} + bn^{2} + cn + d\\) is also an integer, find the minimal value of \\(a\\). The original answer is in \\(\\frac{k}{m}\\) format, please give the value of k + m.\n\nRemember to place... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
[
{
"content": "What is the slope of the line passing through the points (6, 8) and (8, 6)?\n\nRemember to place the final answer in the last part using the format: \n\n\\boxed{{'The final answer goes here.'}}",
"role": "user",
"tool_calls": null
},
{
"content": "<think>\nI need to find the slope ... | [
{
"function": {
"description": "A tool for executing code.",
"name": "code_interpreter",
"parameters": {
"properties": {
"code": {
"description": "The code to execute.",
"type": "string"
}
},
"required": [
"code"
... |
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