prompt_messages listlengths 4 4 | answers sequencelengths 4 4 | gt stringlengths 1 44 | rewards sequencelengths 4 4 |
|---|---|---|---|
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Square $ABCD$ has side length 2, and $X$ is a point outside the square such that $AX = XB = \\sqrt{2}$. What is the length of the lo... | [
"[VERIFY] correct.\n\nThe steps and reasoning provided are correct. The placement of the square and the coordinates of point \\( X \\) are accurate. The distances calculated using the distance formula are correct, and the determination of the longest diagonal is accurate.",
"[VERIFY] correct.\n\nThe steps are as ... | \sqrt{\sqrt{10}} | [
-1,
-1,
-1,
-0.5
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ S = \\{1, 2, \\cdots, n\\} $. Define $ A $ as a collection of some subsets of $ S $ such that no two elements in $ A $ are mut... | [
"[VERIFY] correct.\n\nThe reasoning and application of Sperner's Theorem are correct. The theorem directly applies to the problem, and the calculation of the maximum number of elements in the collection \\(A\\) using the binomial coefficient \\( \\binom{n}{\\left\\lfloor \\frac{n}{2} \\right\\rfloor} \\) is accurat... | C_n^{\left\lfloor \frac{n}{2} \right\rfloor} | [
-1,
-0.5,
-0.5,
-0.5
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the result when the expression $\\dfrac{\\sqrt[5]{11}}{\\sqrt[6]{11}}$ is expressed as 11 raised to what power? Let's think ... | [
"[VERIFY] correct.\n\nThe steps follow the proper mathematical rules for handling exponents and radicals. The conversion from radical form to exponential form, the subtraction of exponents, and the simplification of the fraction are all performed accurately.",
"[VERIFY] correct.",
"[VERIFY] correct.\n\nThe step... | \frac{1}{30} | [
-1,
-1,
1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many two-digit prime numbers can be formed by choosing two different digits from the set $\\{3, 5, 7, 9\\}$ to be used as the te... | [
"[VERIFY] correct.\n\nThe steps followed were:\n1. List all possible two-digit numbers formed by the digits from the set \\{3, 5, 7, 9\\}.\n2. Check each number for primality.\n3. Count the prime numbers.\n\nEach step was followed correctly and no mistakes were made in the process.",
"[VERIFY] correct.",
"[VERI... | 6 | [
-1,
-1,
1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given points $M(0,3)$, $N(-4,0)$, and $P(-2,4)$;\n1. Find the equation of line $l$ that passes through point $P$ and is parallel to ... | [
"[VERIFY] correct.\n\nThe steps and calculations are correct. The equation of the line \\( l \\) and the area of \\(\\triangle MNP\\) were derived correctly.",
"[VERIFY] correct.\n\nThe steps and calculations provided are correct. The slope of line $MN$ was correctly calculated, the equation of line $l$ was deriv... | S=5 | [
-1,
-1,
1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A jar is filled with green, blue, red, and yellow jelly beans. The probability of randomly selecting a green jelly bean is 0.25 and ... | [
"[VERIFY] correct.\n\nThe steps taken to determine the probability of randomly selecting either a red or yellow jelly bean were as follows:\n\n1. Identified the probabilities of selecting green and blue jelly beans, which were given as \\(0.25\\) and \\(0.35\\) respectively.\n2. Calculated the combined probability ... | 0.40 | [
1,
-1,
1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ \\triangle ABC $, respectively. Set $ O $ as the origin of the... | [
"[VERIFY] correct.\n\nThe steps taken to derive the complex number corresponding to the orthocenter \\( H \\) of triangle \\( \\triangle ABC \\) are correct. The reasoning follows from the properties of complex numbers in the context of triangle geometry, specifically the fact that the sum of the complex numbers of... | z_{1} + z_{2} + z_{3} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The number of sets A that satisfy A ∪ {-1, 1} = {-1, 0, 1} Let's think step by step and output the final answer within $\boxed{}$.",... | [
"[VERIFY] correct.\n\nThe steps and the reasoning provided are consistent and correctly follow the necessary conditions to find the number of sets \\( A \\) that satisfy \\( A \\cup \\{-1, 1\\} = \\{-1, 0, 1\\} \\). Each step logically follows from the previous one and the final answer is correct.",
"[VERIFY] cor... | 4 | [
-1,
-1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the ratio of $ x $ to $ y $ if: $\\frac{8x - 5y}{11x - 3y} = \\frac{2}{3}$? Express your answer as a common fraction. Let's ... | [
"[VERIFY] correct.\n\nThe steps followed to solve the problem were logical and mathematically sound. Each step was justified and correctly applied algebraic manipulations to isolate the variables and find the ratio \\( \\frac{x}{y} \\). The final ratio \\( \\frac{9}{2} \\) is derived correctly from the equation \\(... | \frac{9}{2} | [
-1,
-1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ S = \\{1, 2, \\cdots, n\\} $. Define $ A $ as a collection of some subsets of $ S $ such that no two elements in $ A $ are mut... | [
"[VERIFY] correct.\n\nThe steps taken to solve the problem are correct and align with Sperner's Theorem. The reasoning and application of the theorem are accurate, and the final answer is properly formatted.",
"[VERIFY] correct.\n\nThe steps followed to derive the solution are correct and adhere to Sperner's Theo... | C_n^{\left\lfloor \frac{n}{2} \right\rfloor} | [
-1,
-0.5,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run... | [
"[VERIFY] correct.\n\nThe solution steps are logically consistent and correctly apply the given relationships between the speeds of Usain, his mum, and Turbo the tortoise. Each step follows from the previous ones accurately, ensuring the final answer is correct.",
"[VERIFY] correct.\n\nThe steps taken to solve th... | 40 \, \text{m} | [
-1,
1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the number of six-digit palindromes in base 8. Let's think step by step and output the final answer within $\boxed{}$.",
"r... | [
"[VERIFY] correct.\n\nThe steps for determining the number of six-digit palindromes in base 8 are as follows:\n\n1. Recognize the form of a six-digit palindrome in base 8 is $abc cba$, where $a, b, c \\in \\{0, 1, 2, 3, 4, 5, 6, 7\\}$, and $a \\neq 0$.\n2. Calculate the number of choices for $a$ (1 to 7), which giv... | 3584 | [
-0.5,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ g(x) = 3x - 2 $. Find the sum of all $ x $ that satisfy the equation $ g^{-1}(x) = g(2x) $. Let's think step by step and outpu... | [
"[VERIFY] correct.\n\nThe solution follows a logical and correct step-by-step process to find the inverse function, substitute it into the given equation, and solve for \\( x \\). There are no errors in the justifications and calculations.",
"[VERIFY] correct.\n\nThe steps followed to find the solution are correc... | \frac{8}{17} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ \\triangle ABC $, respectively. Set $ O $ as the origin of the... | [
"[VERIFY] correct.\n\nThe derivation follows from the fundamental property of the orthocenter in the complex plane when the circumcenter is at the origin. The orthocenter \\(H\\) is given by the sum of the complex numbers representing the vertices of the triangle, which is mathematically sound. There are no incorre... | z_{1} + z_{2} + z_{3} | [
-1,
-1,
-1,
-0.5
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "When mailing a letter at the post office, the postage is 0.8 yuan for a letter weighing no more than 20g, 1.6 yuan for a letter weig... | [
"[VERIFY] correct.\n\nThe steps to determine the postage for a 78.5 gram letter were followed correctly:\n1. Identified the base postage and weight intervals.\n2. Calculated the number of 20-gram intervals correctly.\n3. Calculated the additional postage correctly.\n4. Added the base postage and additional postage ... | 3.2 | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate: $\\left(-2b\\right)^{3}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within $\boxed{}$.",
... | [
"[VERIFY] correct.\n\nThe steps are as follows:\n1. The expression $\\left(-2b\\right)^{3}$ means multiplying $-2b$ by itself three times.\n2. First, we calculate $\\left(-2b\\right) \\times \\left(-2b\\right) = 4b^2$.\n3. Then, we multiply $4b^2$ by $-2b$ to get $-8b^3$.\n\nEach step is logically sound and the cal... | -8b^{3} | [
-1,
1,
1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate: $\\left(-2b\\right)^{3}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within $\boxed{}$.",
... | [
"[VERIFY] correct.\n\nThe steps followed to calculate \\(\\left(-2b\\right)^{3}\\) were correct. We expanded the expression to \\((-2b) \\times (-2b) \\times (-2b)\\), multiplied the constants, and then the variables, and finally combined the results. Each step was logically and mathematically sound.",
"[VERIFY] ... | -8b^{3} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate: $\\left(-2b\\right)^{3}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within $\boxed{}$.",
... | [
"[VERIFY] correct.\n\nThe steps followed to evaluate $\\left(-2b\\right)^{3}$ are correct. Each step logically follows from the previous one, correctly applying the rules of exponents and multiplication. Specifically:\n\n1. Recognizing that $\\left(-2b\\right)^{3} = (-2b) \\times (-2b) \\times (-2b)$.\n2. Multiplyi... | -8b^{3} | [
1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A cylindrical can has a circumference of 18 inches and a height of 8 inches. A spiral strip is painted on it that makes exactly two ... | [
"[VERIFY] correct.\n\nThe steps and calculations are consistent and correct. Each step logically follows from the previous one, and the final answer is derived correctly from the given information and geometric principles.",
"[VERIFY] correct.",
"[VERIFY] correct.\n\nThe steps followed correctly account for the... | \sqrt{1552} \text{ inches} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the result when the expression $\\dfrac{\\sqrt[5]{11}}{\\sqrt[6]{11}}$ is expressed as 11 raised to what power? Let's think ... | [
"[VERIFY] correct.\n\nThe steps are consistent with the rules of exponents and the operations performed are correct. Specifically:\n- Rewriting the radicals as exponents is correct.\n- Applying the exponent rule \\(\\dfrac{a^m}{a^n} = a^{m-n}\\) is correct.\n- Subtracting the exponents \\(\\dfrac{1}{5} - \\dfrac{1}... | \frac{1}{30} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the result when the expression $\\dfrac{\\sqrt[5]{11}}{\\sqrt[6]{11}}$ is expressed as 11 raised to what power? Let's think ... | [
"[VERIFY] correct.\n\nThe solution follows the steps of rewriting radicals as exponents, using the properties of exponents for division, finding a common denominator to subtract the exponents, and simplifying the expression correctly. Each step is logically sound and leads to the correct final answer.",
"[VERIFY]... | \frac{1}{30} | [
-1,
-1,
1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ \\triangle ABC $, respectively. Set $ O $ as the origin of the... | [
"[VERIFY] correct.\n\nThe derivation of the complex number corresponding to the orthocenter \\( H \\) of triangle \\( \\triangle ABC \\) with the circumcenter \\( O \\) as the origin is correct. Each step logically follows from the previous one, and the final expression \\( H = a + b + c \\) accurately represents t... | z_{1} + z_{2} + z_{3} | [
-1,
-1,
-0.5,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the number of six-digit palindromes in base 8. Let's think step by step and output the final answer within $\boxed{}$.",
"r... | [
"[VERIFY] correct.\n\nThe reasoning:\n1. A six-digit palindrome in base 8 is of the form $abc cba$. This means the first and last digits are the same, the second and second-to-last digits are the same, and the third digit is the same as middle digit.\n2. The first digit $a$ can be any digit from 1 to 7 (7 choices),... | 3584 | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A jar is filled with green, blue, red, and yellow jelly beans. The probability of randomly selecting a green jelly bean is 0.25 and ... | [
"[VERIFY] correct.\n\nIn the solution provided:\n1. We correctly used the fact that the sum of probabilities for all possible outcomes must equal 1.\n2. We correctly substituted the given probabilities for green and blue jelly beans.\n3. We correctly combined the known probabilities and isolated the sum of the prob... | 0.40 | [
-1,
-1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the ratio of $ x $ to $ y $ if: $\\frac{8x - 5y}{11x - 3y} = \\frac{2}{3}$? Express your answer as a common fraction. Let's ... | [
"[VERIFY] correct.\n\nThe steps followed to solve the equation and find the ratio \\(\\frac{x}{y}\\) are correct. Each step was logically derived and verified through algebraic manipulation, ensuring the solution is accurate.",
"[VERIFY] correct.",
"[VERIFY] correct.",
"[VERIFY] correct.\n\nThe steps followed... | \frac{9}{2} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the result when the expression $\\dfrac{\\sqrt[5]{11}}{\\sqrt[6]{11}}$ is expressed as 11 raised to what power? Let's think ... | [
"[VERIFY] correct.\n\nThe steps are as follows:\n\n1. Rewriting the radicals as exponents: $\\sqrt[5]{11} = 11^{1/5}$ and $\\sqrt[6]{11} = 11^{1/6}$. Therefore, $\\frac{\\sqrt[5]{11}}{\\sqrt[6]{11}} = \\frac{11^{1/5}}{11^{1/6}}$.\n2. Applying the property of exponents for division: $\\frac{11^{1/5}}{11^{1/6}} = 11^... | \frac{1}{30} | [
-1,
-1,
-1,
-1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Simplify the expression: $$\\sqrt[3]{1728000}$$ Let's think step by step and output the final answer within $\boxed{}$.",
"role"... | [
"[VERIFY] correct.",
"[VERIFY] correct.\n\nThe steps are as follows:\n1. Factorize 1728000 into its prime factors:\n $1728000 = 2^9 \\times 3^3 \\times 5^3$.\n2. Apply the cube root to the prime factorization:\n $\\sqrt[3]{1728000} = \\sqrt[3]{2^9 \\times 3^3 \\times 5^3} = 2^3 \\times 3^1 \\times 5^1 = 8 \\t... | 120 | [
-1,
1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A jar is filled with green, blue, red, and yellow jelly beans. The probability of randomly selecting a green jelly bean is 0.25 and ... | [
"[VERIFY] correct.\n\nThe steps are as follows:\n\n1. The given probabilities of selecting a green and blue jelly bean are correct.\n2. The sum of all probabilities for the jelly beans in the jar should equal 1.\n3. We subtracted the known probabilities (green and blue) from 1 to find the combined probability of se... | 0.40 | [
1,
-1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many two-digit prime numbers can be formed by choosing two different digits from the set $\\{3, 5, 7, 9\\}$ to be used as the te... | [
"[VERIFY] correct.\n\nThe steps to identify prime numbers and the list of prime numbers are accurate. Each number was checked for divisibility, and only the numbers with exactly two distinct positive divisors (1 and itself) were considered prime. The final count of prime numbers is 6, which is correct based on the ... | 6 | [
-1,
1,
1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Consider the expression:\n$\n\\left(\\frac{x^3+1}{x}\\right)\\left(\\frac{y^3+1}{y}\\right) + \\left(\\frac{x^3-1}{y}\\right)\\left(... | [
"[VERIFY] correct.\n\nThe steps taken to simplify the given expression are correct. Let's go through the steps to ensure clarity and correctness:\n\n1. **Expand each fraction:**\n \\[\n \\left(\\frac{x^3+1}{x}\\right)\\left(\\frac{y^3+1}{y}\\right) = \\frac{(x^3+1)(y^3+1)}{xy},\n \\]\n \\[\n \\left(\\frac... | 2xy^2 + \frac{2}{xy} | [
-1,
-1,
-1,
-0.5
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Simplify the expression: $$\\sqrt[3]{1728000}$$ Let's think step by step and output the final answer within $\boxed{}$.",
"role"... | [
"[VERIFY] correct.\n\nThe steps followed correctly factorized \\(1728000\\), simplified the expression by taking the cube root of each prime factor, and calculated the final product accurately.",
"[VERIFY] correct.\n\nThe steps involved prime factorization and simplification of the cube root of the factors. Each ... | 120 | [
-1,
1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $ g(x) = 3x - 2 $. Find the sum of all $ x $ that satisfy the equation $ g^{-1}(x) = g(2x) $. Let's think step by step and outpu... | [
"[VERIFY] correct.\n\nThe steps taken were correct and followed a logical progression to find the solution. The function and its inverse were correctly determined, the equation was set up properly, and solved accurately. The final sum of all $x$ that satisfy the equation is $\\boxed{\\frac{8}{17}}$.",
"[VERIFY] c... | \frac{8}{17} | [
1,
-1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If one root of the quadratic equation $x^{2}-2x+a=0$ is $3$, find the other root. Let's think step by step and output the final answ... | [
"[VERIFY] correct.\n\nThe steps are consistent with the properties of quadratic equations and the given conditions. The sum of the roots was correctly identified and used to find the unknown root.",
"[VERIFY] correct.\n\nThe steps are as follows:\n\n1. For the quadratic equation $x^2 - 2x + a = 0$, the sum of the... | -1 | [
-1,
1,
-1,
1
] |
[
{
"content": "You are a mathematical reasoning assistant. Always ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A jar is filled with green, blue, red, and yellow jelly beans. The probability of randomly selecting a green jelly bean is 0.25 and ... | [
"[VERIFY] correct.\n\nThe steps are as follows:\n1. Sum of probabilities: $ P(R) + P(Y) + P(G) + P(B) = 1 $\n2. Substitute known values: $ P(R) + P(Y) + 0.25 + 0.35 = 1 $\n3. Simplify to find $ P(R) + P(Y) $: $ P(R) + P(Y) = 0.40$\n\nAll steps are logically correct and lead to the final answer of 0.40.",
"[VERIFY... | 0.40 | [
1,
-1,
-1,
-1
] |
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