instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that $\cos x = 2m - 1$, and $x \in \mathbb{R}$, determine the range of values for $m$. Express your answer as an interval in standard interval notation. | [0, 1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each o... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets $S=\{a_1, a_2\}$ and $T=\{b_1, b_2\}$, determine the total number of mappings from set $S$ to set $T$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the original letter arrangement "abcd", how many different rearrangements are there such that no two originally adjacent letters are adjacent in the new arrangement? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of the real number 4. If there are multiple roots, express your answer using the ± symbol. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Positive integers \( s \) and \( t \) have the property that \( s(s - t) = 29 \). Calculate the value of \( s + t \). Express your answer as a single integer. | 57 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \(6102 - 2016\)? Express your answer as a single integer. | 4086 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that 148 students have scores ranging from 100 to 120 (inclusive), with scores being positive integers, and students with the same score are given the same rank, calculate the smallest possible number x such that each rank can have at most x students. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming's mother made several transactions on New Year's Day using her WeChat wallet. The transactions are listed below. Calculate the net result of these transactions, specifying whether it's an income or expense, and provide the amount in 元.
| Transaction | Amount (元) |
|------------------------|-------... | expenses of 39元 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Recycling 1 tonne of paper will save 24 trees. If 4 schools each recycle 3/4 of a tonne of paper, then calculate the total number of trees this will save. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the ratio of the radii of three spheres is 1:2:3, find the multiple of the sum of the surface areas of the two smaller spheres that equals the surface area of the largest sphere. Express your answer as a numerical value. | 9/5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(PQRS\) is a square with side length 8. Points \(T\) and \(U\) are on \(PS\) and \(QR\) respectively with \(QU = TS = 1\). Calculate the length of \(TU\). Express your answer as a numerical value closest to a whole number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Among the courses chosen by two people, A and B, from 4 courses, each choosing two, find the number of ways in which at least one course is different between A and B. Express your answer as a numerical value. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The point M has coordinates (-5,2). What are the coordinates of M after reflecting it across the y-axis? Express your answer as an ordered pair (x, y). | (5,2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{b_n\}$ is a geometric sequence, and $b_9$ is the arithmetic mean of 3 and 5, find the product $b_1b_{17}$. Express your answer as a single number. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial expansion $(1 - 2x)^7 = a_0 + a_1x + a_2x^2 + \dots + a_7x^7$, find the value of $|a_0| + |a_1| + |a_2| + \dots + |a_7|$. Express your answer as a single integer. | 2187 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. Given that $a^{2} = b(b+c)$ and $\angle C$ is an obtuse angle, determine the size relationship between $a, 2b, c$. Express your answer using inequality symbols to denote the relationship between the three quantities. | a < 2b < c | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation cos(2α)/sin(α-π/4) = -√6/2, find the exact value of cos(α-π/4). Express your answer as a single value, including any necessary radicals. | \frac{\sqrt{6}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^2+(a+2)y^2+2ax+a=0$, find the value of $a$ such that the equation represents a circle. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A larger cube has a volume of 64 cm³. A smaller cube has edges that are half the length of the edges of the larger cube. Find the volume of the smaller cube. Express your answer in cubic centimeters (cm³) as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(2x) = log3(8x^2 + 7), find the value of f(1). | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} -x-4, & x\leq -1 \\ x^{2}-5, & x>-1 \end{cases}$, find all real numbers $a$ that satisfy the equation $f(a)-11=0$. Express your answer as a list of real numbers. | -15, 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle, the sides are $a$ and $b$, and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. Given that $a:b = 1:3$, calculate the ratio of $r$ to $s$. Express your answer as a simplified ratio in the form $a:b$. | 1:9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and for $x > 0$, $f(x) = \begin{cases} ax+\log_{5}x, & x>4 \\ x^{2}+2^{x}+3, & 0<x\leq4\end{cases}$. If $f(-5) < f(2)$, then find the range of values for $a$. Express your answer as an interval of real numbers. | (-\infty, 2) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{-1,0,1,2,3\}$ and $B=\{x|0 < x < 3\}$, find the intersection $A\cap B$. Express your answer as a set of elements, listing them explicitly. | \{1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A$ and $B$, define $A \otimes B = \{x | x = m - n, m \in A, n \in B\}$. If $A = \{4, 5, 6\}$ and $B = \{1, 2, 3\}$, then find the sum of all elements in the set $A \otimes B$. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x\in\mathbb{Z}\,|\,|x-1|\leqslant 2\}$ and $B=\{x\,|\,\log_{2}(x-1)\leqslant 1\}$, calculate the number of elements in the set $A\cap B$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
A function f is defined recursively by f(1) = f(2) = 1 and f(n) = f(n-1) - f(n-2) + n for all integers n ≥ 3. What is the value of f(2018)? Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the smaller circle touches the larger circle and also passes through its center. What fraction of the area of the larger circle is outside the smaller circle? Express your answer as a fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the last two digits of the sum
$$
2005 + 2005^{2} + 2005^{3} + \cdots + 2005^{2005}
$$
Express your answer as a two-digit number, including a leading zero if necessary. | 05 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Two numbers are such that their difference, their sum, and their product are to one another as 1:7:24. Find the product of the two numbers. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Four osmanthus trees are transplanted within the campus. Given that the probability of survival for each tree is $\frac{4}{5}$, find the variance of the number of surviving trees $\xi$. Express your answer as a fraction. | \frac{16}{25} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the equation of the directrix for the parabola y = 2x^2. Express your answer in the form y =..., including the correct sign and value. | y=-\dfrac{1}{8} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) is a complex number satisfying \( x^{2} + x + 1 = 0 \), calculate the value of \( x^{49} + x^{50} + x^{51} + x^{52} + x^{53} \). Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \left\{\begin{array}{l}{x-2, x\geq10}\\{f(x+6), x<10}\end{array}\right.$, calculate $f(9)$. Provide your answer as a single integer. | 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{2}y^{2}$ in the expansion of $\left( \frac{x}{ \sqrt{y}}- \frac{y}{ \sqrt{x}}\right)^{8}$. Express your answer as a single integer. | 70 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(1,\sin α) $ and $\overrightarrow{b}=(2,4\cos α) $, if $\overrightarrow{a}//\overrightarrow{b}$, find the value of $\tan α$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x\in \mathbb{N} | x^{2}-5x\leq 0\}$ and $B=\{0,2,5,7\}$, find the intersection $A\cap B$. Express your answer as a set of numbers using standard set notation (e.g., {a, b, c}). | \{0,2,5\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangular prism $ABC-A_1B_1C_1$, $\angle BAC=90°$, and the area of the lateral face $BCC_1B_1$ is 4. Find the minimum surface area of the circumscribed sphere of the right triangular prism $ABC-A_1B_1C_1$. Express your answer as a multiple of $\pi$. | 8\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that (F1, F2) are the two foci of the ellipse x^2/16 + y^2/4 = 1, and P is a point on the ellipse such that PF1 is perpendicular to PF2. Determine the area of triangle PF1F2. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the following four equations, determine how many of them have a graph that does not pass through the origin (0,0). A graph does not pass through the origin if, when x = 0, y is not equal to 0.
\[ y = x^4 + 1 \quad y = x^4 + x \quad y = x^4 + x^2 \quad y = x^4 + x^3 \]
Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)= \begin{cases} \sin x,x\in[0,1] \\ x^{2},x\in[1,2] \end{cases}$, find the value of $\int_{0}^{2}f(x)dx$. Express your answer as a single mathematical expression. | \dfrac {10}{3}-\cos 1 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In trapezium PQRS, the sides PQ and SR are parallel. Angle RSP is 120 degrees and PS = SR = 1/3 PQ. Find the size of angle PQR. Express your answer in degrees as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the units digit of the expression $3^{2012}-1$. Express your answer as a single digit (0-9). | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A student multiplied 66 by the repeating decimal \(1.\overline{ab}\) (where \(a\) and \(b\) are two digits), but the student did not notice the repeating nature and incorrectly multiplied 66 by \(1.ab\). Later, it was found that the incorrect answer was 0.5 less than the correct answer. Calculate the two-digit number \... | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The common difference of the arithmetic sequence $\{a_{n}\}$ is not zero, with the first term $a_{1}=1$. $a_{2}$ is the geometric mean of $a_{1}$ and $a_{5}$. Calculate the sum of the first 9 terms of the sequence $\{a_{n}\}$. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve $f(x) = xe^x$ at the point $(0, f(0))$. Express your answer as a linear equation in the form $y = mx + b$. | y = x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle is divided into three smaller congruent rectangles, and each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the rectangles. Express your answer as a simplified ratio (e.g., x:y). | \sqrt{3} : 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor$ does not hold? Express your answer as a non-negative integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x$ in the expansion of $({{x^2}-\frac{2}{x}})^5$. Express your answer as a single integer. | -80 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let $n$ be a natural number. Calculate the value of the expression $C_{n}^{0}{2}^{n}-C_{n}^{1}{2}^{n-1}+…+{\left(-1\right)}^{k}C_{n}^{k}{2}^{n-k}+…+{\left(-1\right)}^{n}C_{n}^{n}$ using the binomial theorem. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Combinatorics -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax + b, and its composition f(f(f(x))) = 27x - 52, calculate the function g(x) such that g(f(x)) = x. Express your answer as a function of x, using function notation g(x). | g(x) = \frac{1}{3}x + \frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, x)$, $\overrightarrow{b} = (2-3x, 2)$, $\overrightarrow{c} = (-1, 2)$, if $\overrightarrow{a}$ is perpendicular to ($\overrightarrow{b} + \overrightarrow{c}$), calculate the value of $x$ that satisfies this condition. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle, one of the acute angles is $40^{\circ}$. Find the measure of the other acute angle in the right triangle. Express your answer in degrees as a whole number. | 50 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (2-ax)^6, find the value of the real number a such that the coefficient of the x^3 term in the expansion of f(x) equals -160. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(x) = -1/4 and x ∈ (π, 3π/2), calculate the value of x in radians. Express your answer using the arcsin function and include π where necessary. | x = π + arcsin(1/4) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x|-2<x<1\}$ and set $B=\{x|0<x<2\}$, find the intersection of set $A$ and $B$. Express your answer in interval notation. | \{x|0<x<1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where each term is positive, if $a_5a_6=81$, calculate the value of $\log_{3}a_1+\log_{3}a_2+\ldots+\log_{3}a_{10}$. Express your answer as a single numerical value. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Point P is a moving point on the parabola $y^2=4x$. The distance from point P to the line $x=-1$ is 4. Find the distance from P to the focus of the parabola $y^2=4x$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( c \) is a real number and the negative of one root of \( x^{2} - 3x + c = 0 \) is a root of \( x^{2} + 3x - c = 0 \), calculate the roots of \( x^{2} - 3x + c = 0 \). Express your answer as two distinct roots. | 0, 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A phenotypically normal couple has a son who has both hemophilia (a sex-linked recessive trait) and phenylketonuria (an autosomal recessive trait). If they have another daughter, calculate the probability that she is phenotypically normal. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation: $\frac{x}{x-1}+\frac{2}{1-x}=2$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Milly adds up all the integers from 1 to n inclusive, and Billy adds up all the integers from n+1 to 20 inclusive. Their totals are the same. Find the value of n that makes the two sums equal. Express your answer as a single positive integer. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of $x^2$ in the expansion of $(1+x)^{3}+(1+x)^{4}+\cdots +(1+x)^{8}$. Express your answer as a single integer. | 83 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinate equation of the hyperbola that shares a common focus with the ellipse $\frac{y^{2}}{49} + \frac{x^{2}}{24} = 1$ and has an eccentricity $e = \frac{5}{4}$. Express your answer in the standard form of a hyperbola equation. | \frac{y^{2}}{16} - \frac{x^{2}}{9} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -x^2 - cos(x), find the solution set of the inequality f(x-1) > f(-1). Express your answer in interval notation. | (0,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = log3(x+1), find the value of x that satisfies the equation f^(-1)(x) = 8. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x\geqslant 5$, find the minimum value of the function $f(x)= \frac {x^{2}-4x+9}{x-4}$. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\cos\left(-\dfrac{23}{4}π\right)$. Express your answer as a fraction or radical, using any necessary mathematical notation (e.g., square root). | \dfrac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality $ax - b < 0$ is $(1, +\infty)$, find the solution set of the inequality $(ax + b)(x - 3) > 0$. Express your answer as an interval or union of intervals. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(x^2-4)(x^2-1)=(x^2+3x+2)(x^2-8x+7)$, find the number of distinct x values that satisfy the equation. Express your answer as a whole number count. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a= \int_{0}^{2} (1-2x) \, dx$, then calculate the constant term of the binomial $(\frac{1}{2}x^2+ \frac{a}{x})^6$. Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( n \) be any positive integer. Consider a sequence containing only the letters \( \mathrm{A} \) and \( \mathrm{B} \), where at most three consecutive \( \mathrm{A} \)s and at most three consecutive \( \mathrm{B} \)s are allowed. Let \( S(n) \) represent the number of such sequences of length \( n \). Find the rem... | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the ${x}^{2}$ term in the expansion of ${(1+x)}^{2}+{(1+x)}^{3}+\cdots+{(1+x)}^{9}$. Express your answer as a single numerical value. | 120 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A fast food chain sells hamburgers throughout the year. If exactly 25% of the chain's hamburgers are sold in the fall, and the sales in the fall are 4 million, how many million hamburgers are sold in the winter? The sales in the spring and summer are 4.5 million and 5 million, respectively. Express your answer in milli... | 2.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $\dfrac {2x}{3x-1} > 1$ and express the solution set in interval notation. | ( \dfrac {1}{3},1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $100(100-3)-(100\cdot100-3)$. Provide your answer as a single integer. | -297 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola $x^2 = ay$ ($a \in \mathbb{R}$). Express your answer as an ordered pair (x, y). | \left( 0, \frac{a}{4} \right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 - 2i, calculate the value of the expression (z^2 + 3) / (z - 1). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive term arithmetic sequence $\{a_n\}$ that satisfies $a_1 + a_{2014} = 2$, find the minimum value of $\frac{1}{a_{2}} + \frac{1}{a_{2013}}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Miss Spelling, the English teacher, asked five of her students how many of the five of them had done their homework the day before. Daniel said none, Ellen said only one, Cara said exactly two, Zain said exactly three, and Marcus said exactly four. Miss Spelling knew that the students who had not done their homework we... | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
How many ordered pairs of real numbers (x, y) satisfy the following system of equations?
\begin{align*}
x + 3y &= 3 \\
\big||x| - |y|\big| &= 1
\end{align*}
Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 identical computers ready to be distributed to 5 primary schools, among which schools A and B each want at least 2 computers, while the other schools may receive none. Find the total number of different distribution schemes. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{m}=(1,0)$ and $\overrightarrow{n}=(1,1)$, find the value of the real number $k$ such that $\overrightarrow{m}+k\overrightarrow{n}$ is perpendicular to $\overrightarrow{m}$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
If the line $2x-y+2=0$ is parallel to the line $y=kx+1$, then what is the value of the real number $k$? Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. Calculate the time Leo thinks it is. Express your answer in the format HH:MM, including AM/PM des... | 12:30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with positive terms, and $a_{10}a_{11} + a_9a_{12} = 2e^5$, find the value of $\ln a_1 + \ln a_2 + \ldots + \ln a_{20}$. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan(160°) = a, calculate the value of sin(2000°) in terms of 'a'. Express your answer as a fraction with a square root in the denominator. | -\dfrac{a}{\sqrt{1+a^2}} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
An ellipse with the equation $\frac{x^{2}}{5} + \frac{y^{2}}{4} = 1$ has its right focus at point $F$. A line with a slope of $2$ passing through $F$ intersects the ellipse at points $A$ and $B$. If $O$ represents the origin, find the area of triangle $OAB$. Express your answer as a single fraction or decimal value. | \frac{5}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1 + a_7 = 22 and a_4 + a_10 = 40. Find the common difference d. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)= \begin{cases} x-3 \ (x\geqslant 9) \\ f(x+6) \ (x < 9) \end{cases}$, calculate the value of $f(5)$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The diagram is a "map" of Jo's local rail network, where the dots represent stations and the lines are routes. Jo wants to visit all the stations, travelling only by train, starting at any station and ending at any station, with no restrictions on which routes are taken. What is the smallest number of stations that Jo ... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest prime factor of \(106^2 - 15^2\)? Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a basketball game at a middle school, player A scored 1/4 of the total points, player B scored 2/7 of the total points, player C scored 15 points, and the remaining seven players each scored no more than 2 points. What is the total score of the remaining seven players? Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial expansion $(2x+ \sqrt{3})^{100} = a_0 + a_1x + a_2x^2 + \ldots + a_{100}x^{100}$, calculate the value of $(a_0 + a_2 + a_4 + \ldots + a_{100})^2 - (a_1 + a_3 + a_5 + \ldots + a_{99})^2$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the product $(12)(15)(16)$, each factor written in base $b$, equals $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Calculate the value of $s$ in base $b$. Express your answer as a single value in base $b$. | 44 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle $x^2 + y^2 - 2x = 0$, find the coordinates of the center of the circle. Express your answer as an ordered pair (x, y). | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Mary sells machines, selling 1 machine on the first day, 3 machines on the second day, 5 machines on the third day, and so on, with each day selling 2 more machines than the previous day. Calculate the total number of machines she has sold after 20 days. Express your answer as a whole number. | 400 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
An object moves to the left twice in a row, each time by 3 meters. We define moving to the right as positive and moving to the left as negative. Write an algebraic expression that represents the result of these two movements. | -6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given $x, y, z \in \mathbb{R}^*$, satisfying $x - 2y + 3z = 0$, find the minimum value of $$\frac {y^{2}}{xz}.$$ | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
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