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"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 ... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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\n\nRemember to put your answer on its own line after \"Answer:\".",
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"content": "Okay, so I need t... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"content": "Find the greatest integer less than $\\sqrt{10} + \\sqrt{80}$.\n\n\nRemember to put your answer on its own line after \"Answer:\".",
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"content": "Okay, so I need to find the greatest integer that's less than the sum of the sq... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
"tool... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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\n\nRemember to put your answer on its own line after \"Answer:\".",
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"content":... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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\n\nRemember to put you... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
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"content": "Okay, so I need to solve for the ... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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(\"$C$\", (... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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{
"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... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
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"content": "Okay, s... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"content": "Given a positive integer $n$ , it can be shown that every complex number of the form $r+si$ , where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $0,1,2,\\cdots,n^2$ as digits. That is, the equation \n$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\\cdots +a_1(-n... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
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"role": "user",
"tool_calls": null
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"content": "Okay, so I need to figur... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"content": "How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an integer.\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
"tool_calls": null
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"content": "Okay, so I need to... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"content": "The roots of the polynomial \\[x^3 - 52x^2 + 581x - k\\]are distinct prime numbers. Find $k.$\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
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"content": "Alright, let's tackle this problem. We need to f... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"content": "Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
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"content": "Okay, so I have this problem h... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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{
"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\n\nRemember to put your answer on its own l... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
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"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\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
"tool_calls": null
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... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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"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\n\nRemember to put your answer on its own line after \"Answer:\".",
... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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, $... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:\... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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\n\nRemember to put your answer on its own line after \"Answer:\".",
"name": null,
"role": "user",
"tool_calls": null
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{
"content": "Okay, so I ... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
[
{
"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 ... | [{"type": "function", "function": {"type": "function", "function": {"name": "execute_python_script", "description": "Execute a python script and return the stdout and stderr", "parameters": {"type": "object", "properties": {"script": {"type": "string", "description": "the python script to execute"}}, "required": ["scri... |
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