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
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|---|---|---|---|---|---|
Given sets $A=\{x\left|\right.2-x \lt 1\}$ and $B=\{x|x^{2}+2x-15 \lt 0\}$, find the intersection of sets A and B, denoted as A ⋂ B. Express your answer as an interval in standard notation. | (1, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a=25000$ is represented in scientific notation as $2.5\times 10^{4}$, calculate the value of $a^{2}$ in scientific notation. Express your answer in the form $\\boxed{coefficient \times 10^{power}}$. | 6.25 \times 10^{8} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest value of the function f(x) = |x| + |(1 - 2013x)/(2013 - x)|, where x ∈ [-1, 1]. Express your answer as a fraction in simplest form. | \frac{1}{2013} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation y = x^2 + 2x + 1 and the output y = 4, solve for all possible values of x. Present your answer(s) as exact value(s), separating multiple solutions with commas if necessary. | -3, 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex fraction: $\frac{2-i}{1+2i}$. Express your answer as a simplified complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is an imaginary unit, if $\frac{a + i}{2 - i}$ is a purely imaginary number, find the absolute value of $\frac{1}{2} + \frac{a + i}{2 - i}$. Express your answer as a decimal or fraction. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A ball is floating on the surface of a lake. After the lake freezes and the ball is removed, a hole with a diameter of 24cm and a depth of 8cm is left on the ice surface. Find the surface area of the ball. Express your answer in terms of π, as a numerical value multiplied by π. | 676\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear regression equation y = bx + 0.16 and the following data points for sales volume (y) and month code (x):
| Month Code x | 1 | 2 | 3 | 4 | 5 |
| --- | --- | --- | --- | --- | --- |
| Sales Volume y (10,000 vehicles) | 0.5 | 0.6 | 1 | 1.4 | 1.5 |
Calculate the value of b. Express your answer as a decim... | 0.28 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. Given that $S_8=30$ and $S_4=7$, calculate the value of $a_4$, the 4th term of the sequence. Express your answer as a simplified fraction. | \dfrac{13}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac{i-5}{1+i}$, where $i$ is the imaginary unit, find the imaginary part of its conjugate $\\bar{z}$. Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The operation defined for ordered pairs of real numbers is: $(a, b) \oplus (c, d) = (ac + bd, ad + bc)$. If for all $(a, b)$, it holds that $(a, b) \oplus (x, y) = (a, b)$, find the ordered pair $(x, y)$. Express your answer as an ordered pair. | (1, 0) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the power function f(x) passes through the point (3, 1/9), find the maximum value of the function g(x) = (x-1)f(x) on the interval [1,3]. Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of the cube root of 64. Express your answer with the ± symbol if applicable. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given a game where two students, A and B, take turns drawing balls without replacement, with each draw being between 1 and 3 balls, and the person who draws the last ball wins, find a value of n (the total number of balls) for which A has a winning strategy, assuming A goes first. Express your answer as a single value ... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If the polynomial $P = a^2 + 4a + 2014$, find the minimum value of $P$. Express your answer as a single integer. | 2010 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(3,0)$, find the sum $\overrightarrow{a}+\overrightarrow{b}$. Express your answer as a vector in the form (x, y). | (4, 2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
You can choose four positive integers X, Y, Z, and W. What is the maximum number of odd sums that can be obtained from the six sums X+Y, X+Z, X+W, Y+Z, Y+W, and Z+W? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $\frac {1+i}{1-i} \cdot z = 3+4i$, calculate the conjugate of $z$ in the form $a+bi$, where $a$ and $b$ are real numbers. | 4+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the simultaneous equations 2x^2 + 6x + 5y + 1 = 0 and 2x + y + 3 = 0, find the quadratic equation that y satisfies. Express your answer in the form ay^2 + by + c = 0. | y^2 + 10y - 7 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If $|x| = 3x + 1$, then calculate the value of $(4x + 2)^{2005}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if the complex number $z=\frac{i}{1+i}$, calculate the conjugate of $z$ in the form $a + bi$, where $a$ and $b$ are real numbers. | \frac{1}{2}-\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
x and y are both positive numbers and are inversely proportional. When x increases by a%, y decreases by b%. Express the value of b in terms of a. | \frac{100a}{100+a} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola y^2 = 2px with focus F, and points A, B, and C lying on the parabola. If the coordinate of point A is (1, 2) and point F is precisely the centroid of triangle ABC, find the equation of line BC in the standard form Ax + By + C = 0. | 2x + y - 1 = 0 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $$f(x)= \begin{cases} -x & x\leq-1 \\ x^{2} & x>-1\end{cases}.$$ Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value range of the function $f(x) = \begin{cases} 2x, & 0 \leq x \leq 1 \\ 2, & 1 < x < 2 \\ 3, & 2 \leq x \end{cases}$. Express your answer as an interval or a union of intervals. | [0, 2] \cup \{3\} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A variety show on a certain TV station arranges the performance order of six different programs (A, B, C, D, E, F). The first program can only be A or B, and the last program cannot be A. Find the total number of different arrangement methods. Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$ with $f(-1)=\frac{1}{2}$ and $f(x+2)=f(x)+2$, calculate the value of $f(3)$. Express your answer as a single value. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
A comprehensive high school has 3500 high school students and 1500 junior high school students. To understand the students' learning situation, a sample of size $n$ is drawn from the students of this school using stratified sampling. It is known that 70 high school students are drawn. Calculate the total sample size $n... | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a > 0$, $b > 0$. If $\sqrt{3}$ is the geometric mean of $3^{a}$ and $3^{b}$, calculate the minimum value of $\frac{1}{a}+\frac{1}{b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
If vector $\overrightarrow{a}=(2,3)$ and vector $\overrightarrow{b}=(-1,2)$, then calculate the coordinates of $\overrightarrow{a} + \overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (1, 5) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
The 12-tone equal temperament system involves inserting 11 positive numbers between 1 and 2 to form an increasing geometric sequence of 13 numbers. Using the properties of geometric sequences, calculate the fourth inserted number in this sequence. Express your answer as a power of 2, including the exponent. | 2^{\frac{1}{3}} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that in the decimal system we have 10 digits, ranging from 0 to 9, and the carryover rule is "carry one over every ten," as in $47+56=103$. It follows that in the octal system we have 8 digits, ranging from 0 to 7, with the carryover rule as "carry one over every eight." Calculate the following in the octal syste... | 125_8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains. Calculate the number of tetrads (groups of four chromatids formed during the prophase of meiosis I by the two homologous chromosomes, each having two sister chromatids) that can be produced by this c... | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(1,1)$, $b=(2,5)$, and $c=(3,x)$, find the value of $x$ that satisfies the condition $(8a-b) \cdot c=30$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A university uses a systematic sampling method to select a sample of 50 students from 2000 students with identification numbers ranging from 0001 to 2000. The sampled identification numbers are arranged in ascending order. Given that the identification number of the first sampled student is 0003, calculate the identifi... | 1963 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows two congruent regular pentagons and a triangle. The angles marked \( x^\circ \) are equal. Find the measure of angle \( x \) in degrees. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(PQRS\) is a rectangle with \(SR = 15\). Point \(T\) is above \(PS\) and point \(U\) is on \(PS\) so that \(TU\) is perpendicular to \(PS\). If \(PT = 10\) and \(US = 4\) and the area of \(PQRS\) is 180, what is the area of \(\triangle PTS\)? Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(\frac{a+b}{a-b} = 3\), find the ratio \(\frac{a}{b}\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 78 and 36 using the Euclidean algorithm. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let $(2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5$. Calculate the value of $\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}+a_{5}}$. Express your answer as a simplified fraction. | -\frac{122}{121} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A merchant reduces the purchase price by 8%, while keeping the selling price unchanged, resulting in an increase in profit margin from x% to (x+10)%. Calculate the initial profit margin x. Express your answer as a percentage. | 15% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A, B, and C are hiking and want to know the distance to the nearest town. A says, "At least 6 miles." B says, "At most 5 miles." C says, "At most 4 miles." In fact, all three of them are wrong. Let d represent their distance to the nearest town. Determine the interval that d belongs to. Express your answer in interval ... | (5, 6) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $\sin(x+\frac{\pi}{2})$. Express your answer as a trigonometric function of x. | \cos x | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A high school senior is preparing to take the independent college entrance exams and needs to choose 3 out of 6 colleges to apply to. Among these colleges, two have exams scheduled at the same time. Calculate the number of different ways the student can register for the exams. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $t \in \mathbb{R}$, $i$ is the imaginary unit, and complex numbers $z_{1} = 3 + 4i$, $z_{2} = t + i$, and $z_{1} \cdot z_{2}$ is a real number, find the value of $t$. Express your answer as a single real number. | -\dfrac{3}{4} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A chess tournament is planned for teams, each of which has exactly three players. Each player in a team will play exactly once against each player from all the other teams. Given that no more than 250 games can be played in total, what is the maximum number of teams that can enter the tournament? Express your answer as... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2sin(ωx + φ) (ω > 0, 0 < φ < π) satisfies f(-x) = f(x), and its graph intersects with the line y = 2 at two points whose x-coordinates are x1 and x2, respectively, and the minimum value of |x1 - x2| is π, find the values of ω and φ. Express your answer as a pair of values (ω, φ), where ω is a ... | (2, π/2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that a complex number $z$ satisfies $z(1+i)=-2i$, find the value of $z$. Express your answer as a complex number in the form $a+bi$. | -1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the equation 2(a_1+a_4+a_7) + 3(a_9+a_{11}) = 24 holds. Find the value of S_{13} + 2a_7. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = -2x + 5, identify the quadrant of the Cartesian coordinate system through which the graph of this function does not pass. Provide your answer in the format 'X quadrant', where X is the number of the quadrant (e.g., 'Third quadrant'). | Third quadrant | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} - \sqrt {x}, & x > 0 \\ (x- \dfrac {1}{x})^{4}, & x < 0\end{cases}$, calculate the value of $f(f(2))$. Express your answer as a single number. | \dfrac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.000000005 meters in scientific notation. Include units in your answer. | 5 \times 10^{-9} \text{ meters} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function composition $f(e^x) = x$, calculate the value of $f(5)$. Express your answer as a single value. | \ln 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function $f(x)= \frac {1}{2}\sin 2x+ \frac {1}{2}\tan \frac {\pi}{3}\cos 2x$. Express your answer in terms of pi (π). | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( N=\left(7^{p+4}\right)\left(5^{q}\right)\left(2^{3}\right) \) be a perfect cube, where \( p \) and \( q \) are positive integers. Find the smallest possible value of \( p+q \) such that \( N \) is a perfect cube. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality (x-1)/x >= 2. Express your answer in interval notation. | [-1,0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( x \) and \( y \) be positive real numbers. What is the smallest possible value of the expression \( \\frac{16}{x} + \\frac{108}{y} + xy \)? Express your answer as a single number. | 36 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $a=(1,2)$ and $b=(-3,2)$, if $ka+b$ is perpendicular to $a$, then find the value of the real number $k$. Express your answer as a single real number. | -\dfrac{1}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality x^2 - 2ax + a^2 - ab + 4 ≤ 0 has exactly one solution, find the minimum value of a^2 + b^2. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $X \sim N(0, \sigma^2)$ and $P(-2 \leq X < 0) = 0.4$, calculate the probability $P(X > 2)$. Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Two different prime numbers sum up to 10. Find the product of these two prime numbers. Express your answer as a whole number. | 21 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows a cuboid in which the area of the shaded face is one-quarter of the area of each of the two visible unshaded faces. The total surface area of the cuboid is 72 cm². Find the area of one of the visible unshaded faces of the cuboid. Express your answer in square centimeters (cm²). | 16 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: \[ \frac {(1+i)^{2}}{(1-i)^{3}} \] and express your answer in standard form (a + bi). | - \frac {1}{2}- \frac {i}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let the three-digit number n = 100a + 10b + c, where a, b, c ∈ {1, 2, 3, 4} can form the sides of an isosceles (including equilateral) triangle. Count the number of such three-digit numbers n. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle, the ratio of the base angle to the vertex angle is 2:1. Find the degree of the vertex angle. Express your answer in degrees as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference of 2, if a_1, a_2, a_5 form a geometric sequence, find the value of a_2. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 students and 2 teachers standing in a row for a group photo. Find the number of ways they can stand such that the 2 teachers are not adjacent to each other. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x(x-1) < 2$ and express the solution set in interval notation. | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Person A can complete a job in $a$ days, and person B can complete the same job in $b$ days. What is the combined work rate of A and B working together in one day? Express your answer as a mathematical expression involving $a$ and $b$. | \frac{1}{a}+\frac{1}{b} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a scalene triangle $ABC$, two of the altitudes have lengths $4$ and $12$. If the length of the third altitude is also an integer, what is the largest possible value it can have? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangular pyramid S-ABC with its three lateral edges mutually perpendicular, and SA=2, SB=SC=4, find the radius of the circumscribed sphere of this pyramid. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Three boxes under my stairs contain apples or pears or both. Each box contains the same number of pieces of fruit. The first box contains all twelve of the apples and one-ninth of the pears. Calculate the total number of pieces of fruit in each box. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Five equilateral triangles with a side length of $2\sqrt{3}$ are arranged such that the midpoint of the base of each triangle is the vertex of the next triangle. Calculate the total area covered by these triangles in the plane. Express your answer in the format of a numerical value multiplied by $\sqrt{3}$. | 12\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with integer side lengths and a perimeter of 8, find its area. Express your answer as a numerical value, possibly involving a square root. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0, b > 1$, and $a + b = 2$, find the minimum value of $\frac{4}{a} + \frac{1}{b-1}$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A prime number q is called a P-type prime number if q+1 is a perfect square. How many P-type prime numbers exist? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left( heta - \frac{\pi}{4}
ight) = \frac{1}{5}$, calculate the value of $\cos\left( heta + \frac{\pi}{4}
ight)$. Express your answer as a fraction, including the negative sign if applicable. | -\frac{1}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and the set $$M=\{z|z=( \frac {i-1}{i+1})^{n}，n\in\mathbb{N}^{*}\}$$, calculate the number of non-empty subsets of set $M$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A triathlon consists of swimming, cycling, and running. The cycling accounts for three-quarters of the total distance, the running for one-fifth, and the swimming for 2 km. Find the total distance of this triathlon. Express your answer in kilometers as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, given points A(-1,0,1) and B(5,2,1), calculate the coordinates of the midpoint of the line segment AB. Express your answer in the format (x, y, z). | (2, 1, 1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The hyperbola \(x^2 - \frac{y^2}{m} = 1\) has one of its foci at \((-3,0)\). Find the value of \(m\). Express your answer as a single value. | 8 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that π < α < 3π/2 and sin(3π/2 + α) = 4/5, find the value of tan(α/2). Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of bacteria can reproduce to twice its original number after 60 minutes of cultivation. If the initial count of bacteria is 10, calculate the number of bacteria that can be reached after 7 hours of cultivation. Express your answer as a whole number. | 1280 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of a hydrogen atom is 0.0000000001m. Express this value in scientific notation in the form $a\times10^{n}$, where $1\leq|a|<10$, and $n$ is an integer. | 1\times10^{-10} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.75 | 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"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 + a_2 = 3 and a_3 + a_4 = 12. Find the value of a_5 + a_6. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overline{z}$ is the complex conjugate of the complex number $z$, and $(1-i)z=1+i$, calculate the value of $\overline{z}$. Express your answer as a complex number, including the negative sign if applicable. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $2|x-10|+3|x-20| \leq 35$ and express the solution set as an interval, including any relevant brackets or parentheses to denote inclusion or exclusion of endpoints. | [9, 23] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A certain circular part has a diameter requirement of $50\pm 0.2$ mm. The table below shows the results of the hole diameter measurements for 6 produced parts (with a standard value of $50$ mm). How many of these 6 products meet the tolerance requirement?
| Serial Number | 1 | 2 | 3 | 4 | 5 | 6 |
|-... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The marked price of a piece of furniture is 132 yuan. If it is sold at a 10% discount (i.e., 90% of the marked price), and there is still a 10% profit (relative to the purchase price), then what is the purchase price of the furniture? Express your answer in yuan as a whole number. | 108 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given positive integers \(x, y, z\) that satisfy the system of equations:
\[
\left\{
\begin{array}{l}
x^2 + y^2 + z^2 = 2(yz + 1), \\
x + y + z = 4032.
\end{array}
\right.
\]
Calculate the value of \(x^2 y + z\). Express your answer as a single integer. | 4031 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = (-2 + ai) / (1 + i), find the value of the real number a such that z corresponds to a point on the imaginary axis in the complex plane. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Two students, A and B, are traveling to City A. There are three different routes available, and each student has an equal probability of choosing any of the routes. What is the probability that both students choose the same route? Express your answer as a fraction. | \frac {1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( P Q R S \) represents a rectangular piece of paper. The paper is folded along a line \( V W \) so that \(\angle V W Q = 125^{\circ}\). When the folded paper is flattened, points \( R \) and \( Q \) have moved to points \( R^{\prime} \) and \( Q^{\prime} \), respectively, and \( R^{\prime} V \) crosse... | 110^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Among all factors of \(201^9\), how many of them are either perfect squares or perfect cubes? Express your answer as a single integer. | 37 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n}, where a_1 = 1 and a_{n+1} = a_n + 3, if a_n = 2014, find the value of n. Express your answer as a whole number. | 672 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the distance from point P(x, y) to point A(0, 4) is equal to the distance to point B(-2, 0), find the minimum value of $2^x + 4^y$. Express your answer in simplest mathematical notation, including any necessary square roots. | 4\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{4} = 1$ passes through the point (2, -1), calculate the eccentricity of the hyperbola. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function with a period of 2. When 0 <= x <= 1, f(x) = 2x(1-x). Calculate the value of f(-5/2). Express your answer as a fraction. | - \frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
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