conversations listlengths 2 2 | question stringlengths 40 3.04k ⌀ | source stringclasses 5
values | id int64 9.51k 49.6k ⌀ | prompt listlengths 2 2 | prompt_type stringclasses 1
value | model_responses__example_annotation listlengths 2 2 | model_responses__example_annotation__finish_reason_length_flags listlengths 2 2 | model_responses__example_annotation__length_partial_responses listlengths 2 2 | prompt__example_annotation__metadata dict | model_responses__example_annotation__metadata dict |
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"value": "Return your final response within \\boxed{}. Three villages are located in a swampy area. From village $C$, which is equidistant from village $A$ and village $B$, there are straight concrete roads leading to both $A$ and $B$. How can we travel from $A$ to $B$ in the shortest time... | Three villages are located in a swampy area. From village $C$, which is equidistant from village $A$ and village $B$, there are straight concrete roads leading to both $A$ and $B$. How can we travel from $A$ to $B$ in the shortest time if we can travel at speed $v$ on the concrete road and at speed $w$ in the swamp, wh... | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Return your final response with... | original | [
"To determine the optimal path from village \\( A \\) to village \\( B \\) through village \\( C \\), we need to compare the time taken for two possible routes: traveling directly from \\( A \\) to \\( B \\) via the swamp or traveling along the concrete roads.\n\n1. **Direct Path Through Swamp**:\n - The distance... | [
false,
false
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"",
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[
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"value": "Let $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ 2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix}.$ Find the area of the triangle with vertices $\\mathbf{0},$ $\\mathbf{a},$ and $\\mathbf{2b}.$"
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"value": "<think>\nOkay... | null | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Let $\\mathbf{a} = \\begin{pmat... | original | [
"To find the area of the triangle with vertices at $\\mathbf{0}$, $\\mathbf{a}$, and $\\mathbf{2b}$, we can use the formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. The formula is:\n\n\\[\n\\text{Area} = \\frac{1}{2} \\left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_... | [
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"",
""
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"value": "Given $\\cos (\\pi-\\alpha)= \\frac {4}{5}$, and $\\alpha$ is an angle in the third quadrant, the value of $\\tan 2\\alpha$ is ( ).\n\nA: $ \\frac {3}{4}$\n\nB: $- \\frac {3}{4}$\n\nC: $ \\frac {24}{7}$\n\nD: $- \\frac {24}{7}$"
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"value": "<th... | Given $\cos (\pi-\alpha)= \frac {4}{5}$, and $\alpha$ is an angle in the third quadrant, the value of $\tan 2\alpha$ is ( ).
A: $ \frac {3}{4}$
B: $- \frac {3}{4}$
C: $ \frac {24}{7}$
D: $- \frac {24}{7}$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given $\\cos (\\pi-\\alpha)= \\... | original | [
"To find the value of \\(\\tan 2\\alpha\\) given that \\(\\cos (\\pi - \\alpha) = \\frac{4}{5}\\) and \\(\\alpha\\) is in the third quadrant, we will follow these steps:\n\n1. **Simplify \\(\\cos (\\pi - \\alpha)\\):**\n Using the identity \\(\\cos (\\pi - \\alpha) = -\\cos \\alpha\\), we get:\n \\[\n -\\cos ... | [
false,
false
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"",
""
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[
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"value": "Given the descriptions about the function $f(x)=(x^{2}-2x+2)e^{x}- \\frac{e}{3}x^{3}$, the incorrect one is $(\\quad\\quad)$.\nA: No maximum value\nB: The maximum value is $2$\nC: The minimum value is $\\frac{2e}{3}$\nD: The graph of the function $g(x)=f(x)-2$ intersects the $x$-... | Given the descriptions about the function $f(x)=(x^{2}-2x+2)e^{x}- \frac{e}{3}x^{3}$, the incorrect one is $(\quad\quad)$.
A: No maximum value
B: The maximum value is $2$
C: The minimum value is $\frac{2e}{3}$
D: The graph of the function $g(x)=f(x)-2$ intersects the $x$-axis at two points | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given the descriptions about th... | original | [
"To determine which statement is incorrect, we need to analyze the given function \\( f(x) = (x^2 - 2x + 2)e^x - \\frac{e}{3}x^3 \\).\n\nFirst, let's find the critical points by taking the derivative of \\( f(x) \\):\n\n\\[ f'(x) = \\frac{d}{dx}\\left[(x^2 - 2x + 2)e^x - \\frac{e}{3}x^3\\right] \\]\n\nUsing the pro... | [
false,
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"",
""
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"model_name": "hosted_vll... | {
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[
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"value": "Consider the set of all ordered $6$-tuples of nonnegative integers $(a,b,c,d,e,f)$ such that \\[a+2b+6c+30d+210e+2310f=2^{15}.\\] In the tuple with the property that $a+b+c+d+e+f$ is minimized, what is the value of $c$?\n\n\n[i]2021 CCA Math Bonanza Tiebreaker Round #1[/i]"
},
... | Consider the set of all ordered $6$-tuples of nonnegative integers $(a,b,c,d,e,f)$ such that \[a+2b+6c+30d+210e+2310f=2^{15}.\] In the tuple with the property that $a+b+c+d+e+f$ is minimized, what is the value of $c$?
[i]2021 CCA Math Bonanza Tiebreaker Round #1[/i] | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Consider the set of all ordered... | original | [
"To find the tuple \\((a, b, c, d, e, f)\\) that minimizes \\(a + b + c + d + e + f\\) while satisfying the equation \\(a + 2b + 6c + 30d + 210e + 2310f = 2^{15}\\), we need to strategically choose the values of \\(f, e, d, c, b,\\) and \\(a\\).\n\nFirst, note that \\(2^{15} = 32768\\). We start by maximizing the c... | [
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"",
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[
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"value": "Given a sequence $\\{a_n\\}$ where the sum of the first $n$ terms $S_n = 2a_n - 1$, find the general term formula for $a_n$."
},
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"value": "<think>\nOkay, let's see. I have this sequence {a_n} where the sum of the first n terms, S_n, is given by ... | Given a sequence $\{a_n\}$ where the sum of the first $n$ terms $S_n = 2a_n - 1$, find the general term formula for $a_n$. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given a sequence $\\{a_n\\}$ wh... | original | [
"To find the general term formula for \\(a_n\\) given that the sum of the first \\(n\\) terms \\(S_n = 2a_n - 1\\), we start by examining the relationship between consecutive sums.\n\nFirst, let's express \\(S_{n+1}\\) and use the given equation:\n\\[ S_{n+1} = 2a_{n+1} - 1 \\]\nSince \\(S_{n+1} = S_n + a_{n+1}\\),... | [
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"",
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"top_p": 0.8
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
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[
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"value": "Problem 3. How many natural numbers $n>1$ exist, for which there are $n$ consecutive natural numbers, the sum of which is equal to 2016?"
},
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"value": "<think>\nOkay, so I need to figure out how many natural numbers n > 1 exist such that there ar... | Problem 3. How many natural numbers $n>1$ exist, for which there are $n$ consecutive natural numbers, the sum of which is equal to 2016? | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Problem 3. How many natural num... | original | [
"To find how many natural numbers \\( n > 1 \\) exist such that there are \\( n \\) consecutive natural numbers whose sum is 2016, we start by expressing the sum of these \\( n \\) consecutive natural numbers.\n\nLet the first number be \\( a \\). The \\( n \\) consecutive natural numbers can be written as \\( a, a... | [
false,
false
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"",
""
] | {
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"top_p": 0.8
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"model_name": "hosted_vll... | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\\angle{ACE}$ is $90^{\\circ}$. What is the average of the measures, in degrees, of $\\angle{ABC}$ and $\\angle{CDE}$?\n\n[i]2018 CCA Math Bonanza Lightning Round #1.3[/i]"
},
{
"from": "assistant",
... | $ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$?
[i]2018 CCA Math Bonanza Lightning Round #1.3[/i] | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "$ABCDEF$ is a hexagon inscribed... | original | [
"To find the average of the measures of \\(\\angle ABC\\) and \\(\\angle CDE\\), we need to analyze the given information about the hexagon \\(ABCDEF\\) inscribed in a circle with \\(\\angle ACE = 90^\\circ\\).\n\nFirstly, since \\(ABCDEF\\) is a cyclic hexagon, each angle subtended by the same arc will be equal. T... | [
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"",
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"model_name": "hosted_vll... | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "Return your final response within \\boxed{}. If $c$ is a constant such that $x^2+100x+c$ is equal to the square of a binomial, then what is $c$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: If c is a constant such that x² + 100x + c... | If $c$ is a constant such that $x^2+100x+c$ is equal to the square of a binomial, then what is $c$? | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Return your final response with... | original | [
"To find the value of \\( c \\) such that \\( x^2 + 100x + c \\) can be written as the square of a binomial, we start by expressing it in the form \\((x + k)^2\\).\n\nExpanding \\((x + k)^2\\) gives:\n\\[\n(x + k)^2 = x^2 + 2kx + k^2.\n\\]\n\nWe want this to match the given expression \\( x^2 + 100x + c \\). By com... | [
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"",
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"top_p": 0.8
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"model_name": "hosted_vll... | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[{"from":"user","value":"Given an increasing sequence $\\{a_{n}\\}$ where all terms are positive int(...TRUNCATED) | "Given an increasing sequence $\\{a_{n}\\}$ where all terms are positive integers, the sum of the fi(...TRUNCATED) | null | null | [{"content":"You like to solve problems by thinking step by step. Write out your reasoning then give(...TRUNCATED) | original | ["To determine the value of \\( a_n \\) when \\( n \\) takes its maximum value given that \\( S_n = (...TRUNCATED) | [
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