data_source stringclasses 6
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | problem stringlengths 14 2.41k | prompt_messages listlengths 2 2 | gt stringlengths 1 1.47k |
|---|---|---|---|---|---|---|---|
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "p = 3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | p = 3 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3024",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the sequence $\{a_n\}$ is an arithmetic sequence, $a_1=\tan225°$, $a_5=13a_1$, let $S_n$ be the sum of the first $n$ terms of the sequence $\{(-1)^na_n\}$, calculate the value of $S_{2016}$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 3024 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\pm2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the quadratic equation in $x$, $x^2 +kx +1 =0$, has two equal real roots, find the value of $k$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \pm2 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the smallest whole number $b$ such that 80 can be expressed in base $b$ using only three digits? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 5 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=\begin{cases} x+ \frac{2}{x}-3, & x\geqslant 1 \\ \lg (x^{2}+1), & x < 1 \end{cases}$, then $f(f(-3))=$_______. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 0 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "[-2, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the function $f(x) = x^3 - 3x$ has a minimum value on the interval $(a, 10 + 2a^2)$, then the range of real number $a$ is ______. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | [-2, 1) |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "102",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the equation \( a y = b^{2} x^{2} + c \) where \( a, b, c \in \{-3, -2, \cdots, 3\} \) and \( a, b, c \) are distinct from each other, how many different parabolas can be represented by these equations? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 102 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | $E$ and $F$ trisect $\overline{AG}$ and $N$ is the midpoint of $\overline{AG}$. $NF = 10$. What is the length of $\overline{AG}$? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 30 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let
\[\mathbf{C} = \begin{pmatrix} 3 & 2 \\ -8 & -5 \end{pmatrix}.\]
Compute \(\mathbf{C}^{50}\). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "5:7",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In \( \triangle ABC \), point \( D \) is on side \( BC \) such that \( BD = 5 \) and \( DC = 7 \). Determine the ratio of the area of \( \triangle ABD \) to the area of \( \triangle ADC \). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 5:7 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{14}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that
$$(x+y+z)(xy+yz+zx) = 24$$
and
$$x^2(y+z) + y^2(x+z) + z^2(x+y) = 10$$
for real numbers $x$, $y$, and $z$,
what is the value of $xyz$? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \frac{14}{3} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "[\\frac {3}{4}, \\frac {7}{8}]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $f(x)=\sin \omega x-\cos \omega x$ $(\omega > \frac {2}{3})$, if the abscissa of the intersection point of any axis of symmetry of the graph of $f(x)$ with the $x$-axis does not belong to the interval $(\pi,2\pi)$, then the range of $\omega$ is \_\_\_\_\_\_ . (Express the result as an interval) | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | [\frac {3}{4}, \frac {7}{8}] |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1 + \\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Four squares, each with a side length of 2 inches, are placed along a straight line. The second square from the left is lifted out and rotated by 30 degrees. Then it is repositioned centered horizontally between the first and third squares and lowered back until it touches the two adjacent squares. Calculate the vertic... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 + \frac{\sqrt{2}}{2} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "cosβ=- \\frac {56}{65} \\text{ or } cosβ= \\frac {16}{65}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the plane rectangular coordinate system xOy, the vertex of angle α coincides with the coordinate origin, and its initial side coincides with the non-negative semi-axis of the x-axis. Its terminal side passes through point P(-3/5, -4/5).
(1) Find the value of $sin(α+ \frac {π}{3})$;
(2) If angle β satisfies $sin(α+β)... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | cosβ=- \frac {56}{65} \text{ or } cosβ= \frac {16}{65} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "n > 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all natural numbers \( n \) such that the numbers \( 1, 2, 3, \ldots, 2n \) can be partitioned into pairs in such a way that if you sum the numbers in each pair and multiply the sums, the result is the square of a natural number. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | n > 1 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2p^3 + 2p^2 - p + 10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify $(5p^3 - 7p^2 + 3p + 8) + (-3p^3 + 9p^2 - 4p + 2)$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2p^3 + 2p^2 - p + 10 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{2}{{a+1}}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | (1) Factorization: $a^{2}-1$.
(2) Simplify: $\frac{{a-1}}{{{a^2}-1}}+\frac{1}{{a+1}}$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \frac{2}{{a+1}} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "300",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A high school has a total of $900$ students. A stratified sampling method is used to select a sample of $45$ students from the school, with $20$ students from Grade 10, and $10$ students from Grade 12. Determine the number of Grade 11 students in the school. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 300 |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "C) 20:3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The ratio of $w$ to $x$ is $5:2$, of $y$ to $z$ is $3:2$, and of $z$ to $x$ is $1:4$. What is the ratio of $w$ to $y$?
A) 10:3
B) 15:4
C) 20:3
D) 18:3 | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | C) 20:3 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "x^2 + y^2 - 2x + 8y + 9 = 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the standard equation of a circle whose center is on the line $y = -4x$ and that is tangent to the line $l: x + y - 1 = 0$ at the point $P(3, -2)$, and convert it into the general equation of the circle. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | x^2 + y^2 - 2x + 8y + 9 = 0 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the result of $4^{-1}-\sqrt{\frac{1}{16}}+(3-\sqrt{2})^{0}$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the dimensions of a rectangular box with inner dimensions $40 \mathrm{~cm} \times 60 \mathrm{~cm} \times 80 \mathrm{~cm}$ and rectangular wooden blocks with dimensions $20 \mathrm{~cm} \times 30 \mathrm{~cm} \times 40 \mathrm{~cm}$, calculate the maximum number of wooden blocks that could fit into the rectangular... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 8 |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-\\frac{7}{10}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Consider two lines \( L_1 \) and \( L_2 \) with equations \( y = px \) and \( y = qx \) respectively. It is given that \( L_1 \) makes three times the angle with the horizontal x-axis as \( L_2 \) does and that \( L_1 \) has a negative slope that is half the magnitude of the slope of \( L_2 \). If \( q \) is positive a... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | -\frac{7}{10} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{26}{65}, \\frac{16}{64}, \\frac{19}{95}, \\frac{49}{98}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Vanya simplifies fractions by canceling out the same digits in the numerator and denominator. Seryozha noticed that sometimes Vanya gets correct equalities, for example, $49 / 98 = 4 / 8$. Find all correct fractions with numerators and denominators consisting of two non-zero digits that can be simplified in this way. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \frac{26}{65}, \frac{16}{64}, \frac{19}{95}, \frac{49}{98} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\dfrac{5}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate or simplify the following expressions:
(1) $(1)\lg 5(\lg 8+\lg 1000)+{\left(\lg {2}^{ \sqrt{3}}\right)}^{2} $;
(2) Given $a-a^{-1}=2$, find the value of $\dfrac{({{a}^{3}}+{{a}^{-3}})({{a}^{2}}+{{a}^{-2}}-3)}{{{a}^{4}}-{{a}^{-4}}}$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \dfrac{5}{4} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "x-2(x-2)=-3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | When solving the fractional equation $\frac{x}{x-2}-2=\frac{3}{2-x}$, simplify the equation. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | x-2(x-2)=-3 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "665",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let \( A = \{1, 2, 3, \ldots, 1997\} \). For any 999-element subset \( X \) of \( A \), if there exist \( x, y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good subset". Find the largest natural number \( a \) (where \( a \in A \)) such that any 999-element subset containing \( a \) is a... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 665 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A school organized its eighth-grade students to participate in a social practice activity. If they rent several 35-seat buses, they can exactly fill them up. However, if they rent 55-seat buses instead, they can rent one less bus and still have 45 empty seats left.
(1) Find the number of eighth-grade students partici... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1440 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "92",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $S_n$ represents the sum of the first $n$ terms of the sequence ${a_n}$, and $S_{n+1} = S_n + a_n + 3$, $a_4 + a_5 = 23$, determine the value of $S_8$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 92 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(p, q) = (2, 3) \\text{ or } (3, 2)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all prime numbers \( p \) and \( q \) such that the number \( 2^{2} + p^{2} + q^{2} \) is also prime.
Note: A prime number is an integer greater than one that is divisible only by one and itself. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | (p, q) = (2, 3) \text{ or } (3, 2) |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\text{16 pipes of 3-inch diameter and 0 pipes of 5-inch diameter}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A pipe with an inside diameter of 12 inches is used to transport water from a river to a farming community. Assuming no friction or turbulence, find the minimum number of pipes required to carry the same volume of water if you have the option to use either 3-inch or 5-inch inside-diameter pipes. All pipes are of the sa... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \text{16 pipes of 3-inch diameter and 0 pipes of 5-inch diameter} |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-5, -2, -1, 1, 2, 5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A polynomial with integer coefficients is given as:
\[ x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 50. \]
You are informed that the integer \( s \) is a double root of this polynomial. Determine all possible values of \( s \). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | -5, -2, -1, 1, 2, 5 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\tan 105^\\circ = -2 - \\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Compute $\tan 105^\circ.$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \tan 105^\circ = -2 - \sqrt{3} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The number of real roots of the equation $x^3 - 6x^2 + 9x - 10 = 0$ is to be determined. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "69",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | When the product \((4x + 3y + 2)(2x + 5y + 6)\) is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of \(y\)? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 69 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "5y + 23",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify the following expression:
$$7y + 8 - 2y + 15$$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 5y + 23 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "0.3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the probabilities of drawing a red ball and a white ball from a bag are 0.42 and 0.28, respectively, calculate the probability of drawing a black ball from the same bag. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 0.3 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{500}{3}\\pi",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A sphere is perfectly inscribed in a cube. If the edge of the cube measures 10 inches, determine the volume of the sphere in cubic inches. Express your answer in terms of \(\pi\). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \frac{500}{3}\pi |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $a$ is a real number, if $\frac{a-i}{1+i}$ is a pure imaginary number, where $i$ is the imaginary unit, then find the value of $a$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that in a sample, 40 pieces of data are distributed into 4 groups, with the first, second, and fourth groups containing 5, 12, and 8 pieces of data respectively, find the frequency of the third group. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 15 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "7.5 \\text{ m/s} \\text{ or } 12.5 \\text{ m/s}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two cars are driving on two mutually perpendicular roads towards an intersection at constant speeds. Initially, the first car was at a distance of $s_1 = 1600$ meters from the intersection, while the second car was at a distance of $s_2 = 800$ meters from the intersection. The speed of the first car is $v_1 = 72$ km/h.... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 7.5 \text{ m/s} \text{ or } 12.5 \text{ m/s} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | All the positive integers which are co-prime to 2012 are grouped in increasing order in such a way that the \( n^{\text{th}} \) group has \( 2n - 1 \) numbers. For example, the first three groups are (1), (3, 5, 7), (9, 11, 13, 15, 17). It is known that 2013 belongs to the \( k^{\text{th}} \) group. Find the value of \... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 32 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a quadratic equation in $x$, $mx^{2}-(m+2)x+\frac{m}{4}=0$ has two distinct real roots $x_{1}$ and $x_{2}$. If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=4m$, then the value of $m$ is ______. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | **Text**: 7 students are lining up for a photo.
(1) If they are divided into two rows for the photo, with 3 people in the front row and 4 people in the back row, how many different arrangements are there?
(2) If they are arranged into two rows for the photo, with 3 people in the front row and 4 people in the back r... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1440 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Consider polynomials $f(z)$ and $g(z)$, where $f(z)$ has a degree of three and $g(z)$ has a degree of two. The coefficient of the highest degree term of $f(z)$ becomes zero if $z = 1$. What is the degree of $f(z) + g(z)$ when $z = 1$? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | It is known that the cost price of type $A$ disinfectant water is $20$ yuan per bottle, and the cost price of type $B$ disinfectant water is $30$ yuan per bottle. A certain distributor purchases type $A$ and type $B$ disinfectant water for $2000$ yuan for resale (both sales volumes are integers). When the selling price... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 4 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the minimum value of
\[ y = \tan \left( x + \frac{\pi}{3} \right) - \tan \left( x + \frac{\pi}{4} \right) + \sin \left( x + \frac{\pi}{4} \right) \]
for $-\frac{\pi}{2} \le x \le -\frac{\pi}{4}.$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | We have 50 numbered chips from 1 to 50, and they need to be colored either red or blue. We know that chip 5 is blue. For the coloring of the remaining chips, the following rules are followed:
a) If the chip with number $x$ and the chip with number $y$ are of different colors, then the chip with number $|x-y|$ is paint... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Each of the integers \(1, 2, 3, \ldots, 9\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \(\{1, 2, \ldots, 9\}\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 15 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the coordinates of point $A$ are $\left(3,a+3\right)$ and the coordinates of point $B$ are $\left(a,4\right)$, and $AB$ is parallel to the x-axis, calculate the length of segment $AB$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\forall x>0, x^2\\leq0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The negation of the proposition p: $\exists x>0$, $x^2>0$ is $\neg p$: ______. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \forall x>0, x^2\leq0 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "m < \\dfrac {1}{2} \\text{ or } 2 < m\\leqslant \\dfrac {5}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given proposition $p$: The curve $y=x^{2}+(2m-3)x+1$ intersects the $x$-axis at two distinct points; proposition $q$: $\dfrac {x^{2}}{m}+ \dfrac {y^{2}}{2}=1$ represents an ellipse with foci on the $x$-axis. If "$p \lor q$" is a true proposition and "$p \land q$" is a false proposition, find the range of values for $m$... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | m < \dfrac {1}{2} \text{ or } 2 < m\leqslant \dfrac {5}{2} |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the greatest whole number that will satisfy this inequality: \(6x - 5 < 7 - 3x\). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many four-digit numbers with the first digit being 2 have digits that sum to 6? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 15 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A number that is a perfect square is divisible by 3 or leaves a remainder of 1 when divided by 3. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 21 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2x + 5y = 0 \\text{ or } x + 2y + 1 = 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the equation of the line that passes through point A(-5,2) and has an x-intercept that is twice its y-intercept. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2x + 5y = 0 \text{ or } x + 2y + 1 = 0 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(42, 56)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The real number $x$ satisfies $x^2 - 7x + 12 < 0.$ Find all possible values of $x^2 + 7x + 12.$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | (42, 56) |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the real numbers $\dfrac{22}{7}$, $0$, $-\sqrt{2}$, $2\pi$, determine the number of irrational numbers. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "53",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The sum of four consecutive odd integers is 200. What is the largest of these four integers? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 53 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\$60",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Sandy's daughter has a playhouse with one rectangular exterior wall and two triangular faces of the roof, which Sandy plans to cover with a special siding sold in 10-foot by 10-foot sections costing \$30 each. The exterior wall measures 10 feet wide by 8 feet high. Each triangular face of the roof has a base of 10 feet... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \$60 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "35",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let GCF(a, b) be the abbreviation for the greatest common factor of a and b, and let LCM(c, d) be the abbreviation for the least common multiple of c and d. What is GCF(LCM(15, 21), LCM(14, 20))? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 35 |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sqrt{29}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a circle of radius 3 centered at B(3,3) is dilated to a circle of radius 5, and the new center of the circle is at B'(7,10), determine the distance the origin O(0,0) moves under this transformation. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \sqrt{29} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1 \\leq n \\leq 10, n \\in \\mathbb{N}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let S<sub>n</sub> be the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}. Given that S<sub>9</sub> = ‒a<sub>5</sub>.
1. If a<sub>3</sub> = 4, find the general term formula for {a<sub>n</sub>}.
2. If a<sub>1</sub> > 0, find the range of n for which S<sub>n</sub> ≥ a<sub>n</sub>. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 \leq n \leq 10, n \in \mathbb{N} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\left[ -\\frac{\\sqrt{6}}{2}, \\frac{\\sqrt{6}}{2} \\right]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given \( M=\left\{(x, y) \mid x^{2}+2 y^{2}=3\right\} \) and \( N=\{(x, y) \mid y=mx+b\} \). Find the range of values for \( b \) such that \( M \cap N \neq \varnothing \). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \left[ -\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2} \right] |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\left[-1- \\sqrt{2},2- \\sqrt{2}\\right]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given vectors $\overrightarrow{a}=(\cos \omega x-\sin \omega x,\sin \omega x)$ and $\overrightarrow{b}=(-\cos \omega x-\sin \omega x,2\sqrt{3}\cos \omega x)$, let the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}+\lambda$ $(x\in\mathbb{R})$ have its graph symmetric about the line $x=\pi$, where $\omega$ and... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \left[-1- \sqrt{2},2- \sqrt{2}\right] |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3143_5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Convert $423_{10}$ to base 5. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 3143_5 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $a > b > c$, $n \in \mathbb{N}$, and $\frac {1}{a-b} + \frac {1}{b-c} \geq \frac {n^{2}}{a-c}$ always holds. Determine the maximum value of $n$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2 |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "138",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many whole numbers between $200$ and $500$ contain the digit $3$? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 138 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $S$ be the set of 12-tuples $(a_0, a_1, \dots, a_{11})$, where each entry is 0 or 1, so $S$ contains $2^{12}$ 12-tuples. For each 12-tuple $s = (a_0, a_1, \dots, a_{11})$ in $S$, let $p_s(x)$ be the polynomial of degree at most 11 such that
\[p_s(n) = a_n\] for $0 \le n \le 11.$ Find
\[\sum_{s \in S} p_s(12).\] | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2048 |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\binom{N+a+1}{k+1} - \\binom{N}{k+1} = \\sum_{j=0}^{a} \\binom{N+j}{k}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | 1. For all natural numbers \( n \), we have the equality:
$$
\sum_{k=0}^{n}\binom{n}{k}^{2}=\binom{2 n}{n}
$$
2. For all natural numbers \( a \), \( N \), and \( k \), we have:
$$
\binom{N+a+1}{k+1}-\binom{N}{k+1}=\sum_{j=0}^{a}\binom{N+j}{k}
$$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \binom{N+a+1}{k+1} - \binom{N}{k+1} = \sum_{j=0}^{a} \binom{N+j}{k} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "199",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the village, there are 100 houses. What is the maximum number of enclosed, non-intersecting fences that can be built, such that each fence encloses at least one house and no two fences enclose the exact same set of houses? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 199 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For how many values of the digit $A$ is it true that $70$ is divisible by $A$ and $546{,}2A4$ is divisible by 4? | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 1 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the largest of five consecutive two-digit positive integers, all less than 40, which are not prime. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 36 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "4140",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ is a line segment that connects two non-adjacent vertices. Calculate how many diagonals a convex polygon with 25 sides has. Additionally, calculate the sum of all the interior angles in this polygon. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 4140 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "S_{\\triangle ABC_{\\max}} = 2\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, it is known that $\frac{a-c}{a-b}=\frac{\sin(A+C)}{\sin A + \sin C}$.
(I) Determine the size of angle $C$.
(II) Given that $\left|\overrightarrow{CA}-\frac{1}{2}\overrightarrow{CB}\right| = 2$, find the maximum... | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | S_{\triangle ABC_{\max}} = 2\sqrt{3} |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "[-6, 6]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the range of the function \( f \) defined for integers \( k \) by
\[
f(k) = (k)_{3} + (2k)_{5} + (3k)_{7} - 6k
\]
where \( (k)_{2n+1} \) denotes the multiple of \( 2n+1 \) closest to \( k \). | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | [-6, 6] |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\tan C = \\sqrt{6}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $ABC$, $\cot A \cot C = \frac{1}{3}$ and $\cot B \cot C = \frac{2}{9}$. Find $\tan C$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \tan C = \sqrt{6} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "y=\\sin(\\frac{1}{2}x + \\frac{\\pi}{6}), x \\in \\mathbb{R}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Translate the function $y=\sin x$ ($x \in \mathbb{R}$) by moving every point on its graph leftward by $\frac{\pi}{6}$ units, and then stretch the horizontal coordinates of the obtained graph by a factor of 2 (keeping the vertical coordinates unchanged), and determine the function represented by the new graph. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | y=\sin(\frac{1}{2}x + \frac{\pi}{6}), x \in \mathbb{R} |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2! \\times (18 + 6) = 2 \\times 24 = 48",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the number of different arrangements for five people, named A, B, C, D, and E, to stand in a row given that A and B must not stand next to C. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 2! \times (18 + 6) = 2 \times 24 = 48 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\text{The minimum value of } \\sin B - \\sqrt{3}\\sin A \\text{ is } -1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $ABC$, with angles $A$, $B$, and $C$ opposite to sides $a$, $b$, and $c$ respectively, it is known that $\dfrac {a}{c}\cos B + \dfrac {b}{c}\cos A = \dfrac {\sqrt{3}}{2\cos C}$
(I) Find the magnitude of $\angle C$;
(II) Find the minimum value of $\sin B - \sqrt{3}\sin A$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | \text{The minimum value of } \sin B - \sqrt{3}\sin A \text{ is } -1 |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $S$ be a square with vertices labeled $A$, $B$, $C$, and $D$. Determine the number of unique circles in the plane of $S$ that have a diameter with endpoints that are vertices of $S$. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 5 |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(x-1)^{2}+(y+1)^{2}=5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The parabola $y=x^{2}-2x-3$ and the coordinate axes intersect at points that lie on the same circle. The equation of the circle determined by these intersection points is __________. | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | (x-1)^{2}+(y+1)^{2}=5 |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "240",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 8$ and the diameter of the inscribed circle of triangle $OCF$ is 4, compute the product $(AB)(CD).$ | [
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your fi... | 240 |
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