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 $\sqrt[3]{3} \approx 1.442$, calculate the value of $\sqrt[3]{3}-3\sqrt[3]{3}-98\sqrt[3]{3}$. Express your answer as a numerical value, including the negative sign if applicable. | -144.2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integers n < 10^6 such that n is equal to twice the square of an integer and is equal to three times the cube of an integer. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} that satisfies 3a_{n+1} + a_n = 4 for n >= 1 and a_1 = 9, and whose sum of the first n terms is S_n, find the smallest integer n that satisfies the inequality |S_n - n - 6| < 1/125. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A set of n numbers has the sum s. Each number of the set is increased by 20, then multiplied by 5, and then decreased by 20. Find the sum of the numbers in the new set thus obtained. Express your answer in terms of s and n. | 5s + 80n | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let X be a normally distributed random variable with mean μ = 2 and variance σ^2. Given that P(2 < X < 4) = 0.3, find the probability P(0 < X < 4). Express your answer as a numerical value between 0 and 1. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A Buddhist pagoda has seven levels, with the number of red lamps on each level being double that of the level below. If the total number of lamps is 381, how many lamps are there at the top level? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the unit price of a pencil is $a$ yuan, and the unit price of a ballpoint pen is three times the unit price of a pencil, calculate the unit price of a ballpoint pen in terms of $a$. Express your answer as an algebraic expression. | $3a$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows the normal distribution N(0, σ^2), and P(ξ>2) = 0.023, find the probability P(-2≤ξ≤2). Express your answer as a decimal value between 0 and 1, rounded to 3 decimal places. | 0.954 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters. Calculate the total number of different arrangements. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases}1-x,x\leqslant 0 \\ a^{x},x > 0\end{cases}$, find the value of the real number $a$ such that $f(1)=f(-1)$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 3, and 2a_{n+1} - 2a_n = 1. Find the value of a_{99}. Express your answer as a whole number. | 52 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-intercept of the line passing through the points (-1, 1) and (0, 3). Express your answer as a fraction or decimal. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts. Express your answer as a simplified fraction. | \dfrac {3}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
5 volunteers are randomly assigned to 3 different Olympic venues for reception work. Calculate the probability that each venue has at least one volunteer. Express your answer as a simplified fraction. | \frac{50}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle with center at (1, -2) and radius 3. Express your answer in the standard format. | (x-1)^2 + (y+2)^2 = 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function $y=\frac{k}{x}(k≠0)$ and the linear function $y=ax\left(a\neq 0\right)$ intersecting at points $A(-1,2)$ and point $B$, use the symmetry property of the inverse proportion function to find the coordinates of point $B$. Express your answer as coordinates $(x, y)$. | (1,-2) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {-1, 1} and B = {0, 2}, set C is defined as the set of all possible sums of elements from A and B. Calculate the number of proper subsets of set C. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x\log_{3}2 \geq -1$, find the minimum value of the function $f(x) = 4^{x} - 2^{x+1} - 3$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given an algorithm that applies the method of continuous subtraction until the two numbers are equal, determine the number of subtraction operations $i$ required for input values of $a = 252$ and $b = 72$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A class has 6 periods in a day, with 3 periods in the morning and 3 in the afternoon. The schedule includes six subjects: Chinese, Mathematics, English, Physics, Physical Education, and Art. The requirement is that the Mathematics class must be scheduled in the morning and the Art class in the afternoon. Calculate the ... | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an angle in the first quadrant, and $\cos \alpha = \dfrac{5}{13}$, calculate the value of $\dfrac{\sin \left(\alpha - \dfrac{\pi}{4}\right)}{\cos \left(2\alpha + 3\pi \right)}$. Express your answer as a fraction with a square root. | -\dfrac{13\sqrt{2}}{34} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a beehive, there is initially 1 bee. On the first day, it flies out and brings back 5 companions; on the second day, 6 bees fly out, each bringing back 5 companions... If this process of finding companions continues for 6 days, what is the total number of bees in the hive after all the bees have returned on the 6th ... | 46656 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
What is the absolute value of -11? Express your answer as a positive integer. | 11 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Consider the parabola $C$: $y^{2}=4x$ with a focus at $F(1,0)$. A line $l$ with slope $k$ intersects the parabola $C$ at points $M$ and $N$. If the vertical bisector of the line segment $MN$ intersects the $x$-axis at a point with the abscissa $a (a > 0)$, and $n=|MF|+|NF|$, then calculate the value of $2a-n$. Express ... | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar region $\Omega=\{(x, y) \mid x+y \leqslant 6, x \geqslant 0, y \geqslant 0\}$, and another region $A=\{(x, y) \mid x \leqslant 4, y \geqslant 0, x-2 y \geqslant 0\}$, if a point $P$ is randomly chosen within the region $\Omega$, what is the probability that point $P$ falls within region $A$? Express yo... | \frac{2}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the law of motion s=t^2+3, calculate the average velocity of the particle in the time interval (3,3+Δt). Express your answer as a formula including Δt. | 6+Δt | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. Calculate the percentage of flowers that are carnations. Express your answer as a percentage. | 70\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function $f(x)$ defined on $\mathbb{R}$, which satisfies the functional equation $f(x+6) = f(x) + f(3-x)$ for any $x \in \mathbb{R}$, find the value of $f(2010)$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation \(ax + by = 6\) and the conditions \(a, b > 0\), the line forms a triangle with the coordinate axes in the first quadrant. The area of this triangle is 6. Calculate the value of \(ab\). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
There are five teacups and five lids, each numbered from 1 to 5. When covering the teacups with the lids, find the number of ways to have at least two lids match the numbers of their respective teacups. Express your answer as a whole number. | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse y^2/9 + x^2 = 1, a straight line passing through point P (1/2, 1/2) intersects the ellipse at points A and B, and the chord AB is bisected by point P. Calculate the equation of line AB in the standard form Ax + By + C = 0. | 9x + y - 5 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2m,1)$ and $\overrightarrow{b}=(1,-3)$, if $\overrightarrow{a}⊥\overrightarrow{b}$, calculate the real number $m$. Express your answer as a single real number. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integers k for which the equation kx - 12 = 3k has an integer solution for x. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, points B, C, and D lie on a line. Also, ∠ABC = 90° and ∠ACD = 150°. Find the measure of ∠BAC. Express your answer in degrees as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, if a_4 = 4, find the product a_3a_5. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
When rolling two dice simultaneously, what is the probability of obtaining a sum of 9? Express your answer as a simplified fraction. | \dfrac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression: $1-\sqrt{9}$. Provide your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | 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"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an} with a common difference of 2, if a1, a3, a4 form a geometric sequence, find the value of a3. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( AB = 2\sqrt{3} \), \( AC = 2 \), and the height from \( BC \) is \( \sqrt{3} \). Find the length of \( BC \). Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=x+yi$ $(x,y∈R)$, and $\frac{x}{1-i}=1+yi$, $\overline{z}$ is the conjugate of $z$, calculate the imaginary part of $\frac{|z|}{\overline{z}}$. Express your answer as a single value, including any necessary mathematical notation. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\sqrt{1} + \sqrt{9}$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_{n+1} - a_n = 2$ and $a_2 = 5$. Calculate the sum of the first four terms of $\{a_n\}$. Express your answer as a single number. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x|\frac{1}{2}≤{2}^{x+1}≤16\}$ and $B=(-1,4)$, find the intersection $A\cap B$ and express your answer in interval notation. | (-1,3] | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The area of a rectangle is $9a^{2}-6ab+3a$, and one side of the rectangle is $3a$. Calculate the length of the other side. Express your answer as a simplified algebraic expression. | 3a - 2b + 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} (\frac{1}{2})^{x} - 7, & x < 0 \\ x^{2}, & x \geqslant 0 \end{cases}$, find all real numbers $a$ such that $f(a) = 1$. Express your answer as a list of real numbers. | (-3, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} that satisfies 3a_{n+1} + a_n = 4 for n >= 1 and a_1 = 9, and whose sum of the first n terms is S_n, find the smallest integer n that satisfies the inequality |S_n - n - 6| < 1/125. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the points representing $-1$ and $3$ on the number line are points $A$ and $B$ respectively, find the number represented by the point equidistant from point $A$ and point $B$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |x-3| + (y+4)^2 = 0, calculate the value of (x+y)^2023. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
An integer \( n \) is decreased by 2 and then multiplied by 5, resulting in 85. Calculate the exact value of \( n \). | 19 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and additionally, two dresses of different styles. For the May Day performance, she needs to choose one outfit. Calculate the total number of different choices Li Fang has. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The sequences 3, 20, 37, 54, 71,... and 16, 27, 38, 49, 60, 71,... each have 71 as a common term. Find the next term that these sequences have in common. | 258 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m$ and $n$ are roots of the quadratic equation $x^{2}+2x-8=0$, calculate the value of $m^{2}+3m+n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(3, 4), and P(ξ < 2a - 3) = P(ξ > a + 2), find the value of a using the symmetry property of the normal distribution. Express your answer as a single value. | \frac{7}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a linear function y = (m+1)x + (m-1), find the condition on m such that the graph of the function passes through the first, second, and third quadrants. Express your answer as an inequality involving m. | m > 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\frac{3x^2}{\sqrt{1-x}}+\lg (3x+1)$. Express your answer as an interval. | \left(-\frac{1}{3},1\right) | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Calculate this two-digit number. Express your answer as a two-digit number without decimal places. | 28 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Three villages are connected by paths as follows:
1. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path.
2. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path.
3. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfying a_1 = 1/2, a_{n+1} = 3a_n + 1, find the sum S_{2016} of the first 2016 terms of the sequence. Express your answer as a fraction or integer. | \frac{3^{2016}-2017}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A school's mathematics teaching research group uses stratified sampling to survey 35 students from 600 first-year, 780 second-year, and 720 third-year high school students. Calculate the number of second-year students that should be sampled. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-2,-1)$ and $\overrightarrow{b}=(1,2)$, calculate the projection vector $\overrightarrow{c}$ of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$. Express your answer as an ordered pair of fractions. | \left( -\frac{4}{5}, -\frac{8}{5} \right) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function f(x) = xln(x) with domain (0, +∞). Find the minimum value of f(x). Express your answer as a single value in the form \boxed{ }. | -\frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{2} + y^2 = 1$, a line $l$ with an inclination angle of 45° passes through one of the foci of the ellipse and intersects the ellipse at points A and B. Let O be the origin of the coordinates. Calculate the dot product $\overrightarrow {OA} \cdot \overrightarrow {OB}$. | -\frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In a grid with 150 rows and 150 columns, numbered from 1 to 150, the shading pattern is as follows:
- In row 1, every box is shaded.
- In row 2, every second box is shaded.
- In row 3, every third box is shaded.
- This pattern continues such that every $n$th box in row $n$ is shaded.
Find the column number that has th... | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an even function defined on R, and for any x in R, it satisfies f(x + 4) = f(x) + f(2), calculate the exact value of f(2014). | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the triangle \( \triangle ABC \), \( D \) is the midpoint of side \( BC \). Given that \( \overrightarrow{AD} \cdot \overrightarrow{AC} = 0 \), calculate the value of \( \tan A + 2 \tan C \). Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=a^{x-2}+2$ (where $a>0$ and $a \neq 1$), identify a point that is certainly on the graph of this function. Express your answer as an ordered pair (x, y). | (2, 3) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have? Express your answer as a single integer. | 76 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Calculate the value of \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial equations x^3 - 3x^2 + mx + 24 = 0 with roots a, b, and c, and x^2 + nx - 6 = 0 with roots -a and -b, find the value of n. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A car has five tires (four road tires and a full-sized spare) that were rotated so that each tire was used the same number of miles during the first 30,000 miles the car traveled. Calculate the number of miles each tire was used. Express your answer in miles as a whole number. | 24,000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number expression (a+bi)/(4+3i), where i is the imaginary unit, a and b are real numbers, and b is non-zero, find the ratio a/b such that the complex number is purely imaginary. Express your answer as a fraction. | -\frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=\cos ( \frac {1}{2}x- \frac {π}{6})$, translate the function's graph left by $\frac {π}{3}$ units, then halve the x-coordinate of each point on the resulting graph (without changing the y-coordinate). Find the analytical expression of the resulting graph. Express your answer as a single analytical... | y=\cos x | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $sinα= \frac{\sqrt{10}}{10}$, and $α∈(0, \frac{π}{2})$, calculate the value of $cos(2α+ \frac{π}{6})$. Express your answer as a fraction. | \frac{4\sqrt{3}-3}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=xe^{x}$. Express your answer as a single value, including the negative sign if the value is negative. | -\dfrac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
How many coefficients in the binomial expansion of $(x+1)^{65}$ are not divisible by $65$? Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation: $\log_{2} \frac{4}{7} + \log_{2} 7 = x$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $((3i-1)i)$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, if a_1 = 3 and a_5 = 75, then find the value of a_3. Express your answer as a single number. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number z satisfies zi = 2 + i, where i is the imaginary unit, find the modulus of z. Express your answer as a square root. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate Xiao Ming's comprehensive score for the selection of the top three students based on the following weights and scores:
- Average scores of various subjects at the end of the term: 40%
- Weighted class score: 30%
- Discipline and moral scores: 20%
- Practical activities score: 10%
Xiao Ming's scores for the ... | 90.9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of days equivalent to one million seconds (10^6 seconds). Express your answer in days, rounded to the nearest whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that $p$ is a point on the line $x+y-2=0$, and a tangent line is drawn from point $p$ to the circle $O$: $x^{2}+y^{2}=1$ with the points of tangency being $M$ and $N$. If $\angle MPN=90^{\circ}$, determine the number of such points $p$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Three identical squares form rectangle WXYZ, as shown. The perimeter of WXYZ is 56 m. Find the area of WXYZ. Express your answer in square meters (m^2) as a whole number. | 147 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = x + sqrt(2)cos(x) on the interval 0 <= x <= pi/2. Express your answer as a mathematical expression. | \frac{\pi}{4} + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Carl and André are running a race. Carl runs at a constant speed of \( x \) meters per second. André runs at a constant speed of \( y \) meters per second. Carl starts running, and then André starts running 20 seconds later. After André has been running for 10 seconds, he catches up to Carl. Find the ratio \( y : x \) ... | 3 : 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y = (n^2 + n)x^2 - (2n + 1)x + 1, let |A_n B_n| denote the distance between the points A_n and B_n where the parabola intersects the x-axis for each natural number n. Calculate the value of |A_1 B_1| + |A_2 B_2| +... + |A_{1992} B_{1992}|. Express your answer as a fraction in simplest form. | \frac{1992}{1993} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A standard six-faced die is rolled twice. Let $a$ be the number shown on the upper face of the die after the first roll and $b$ be the number shown on the upper face of the die after the second roll. Calculate the probability that the function $y=ax^2-2bx+1$ is a decreasing function on the interval $(-\infty, \frac{1}{... | \frac{5}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A student named Ming participated in a "pull-up" test, where the average number of pull-ups per person was 10. It is known that the first, second, third, and fifth students did 9, 12, 9, and 8 pull-ups respectively. Calculate the number of pull-ups the fourth student did. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 5 boys and 5 girls, select 3 people to form a team to participate in a group project competition, ensuring that there is at least one girl in the team. Calculate the exact number of possible team combinations. | 110 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the tangent line to the curve y = x^3 at the point (1,1), the x-axis, and the line x = 2. Express your answer as a fraction or decimal. | \frac{8}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ and $y$ are positive numbers, find the minimum value of $$(x+ \frac {1}{2y})^{2} + (y+ \frac {1}{2x})^{2}.$$ Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function on $\mathbb{R}$ and satisfies $f(x+2)=-f(x)$, and when $x \in (0, 2)$, $f(x)=2x^2$, calculate the value of $f(2023)$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of the binomial $(x- \frac {1}{2x^{3}})^{8}$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y=x^2-4x+m and two points A(0,y1) and B(1,y2), determine the relationship between y1 and y2. Express your answer using the symbols >, <, or = to describe the relationship between y1 and y2. | y1 > y2 | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Triangles \(XYZ\) and \(PQR\) are drawn on a square grid. Calculate the fraction of the area of triangle \(XYZ\) that is the area of triangle \(PQR\). Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\). | \frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the original function y = x^2 - 1, apply the transformations of shifting it 2 units up and then 1 unit to the right. Express the resulting function in the standard format y = f(x). | y = (x - 1)^2 + 1 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( N = 10^3 + 10^4 + 10^5 + 10^6 + 10^7 + 10^8 + 10^9 \). Calculate the sum of the digits of \( N \). Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the line $x-2y+5=0$ is perpendicular to the line $2x+my-6=0$, find the real number $m$ that makes these lines perpendicular. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A bank teller has some stacks of bills. The total value of the bills in each stack is $1000. Every stack contains at least one $20 bill, at least one $50 bill, and no other types of bills. If no two stacks have the same number of $20 bills, what is the maximum possible number of stacks that the teller could have? Provi... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
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