instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. Calculate the number o... | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the math scores of 40 students in a class for a certain test are x1, x2, x3,..., x40, and the average math score of the whole class is 90. Also, the sum of the squares of the scores is 324400. Calculate the standard deviation of the math scores for this class of students. Express your answer as a mathematica... | \sqrt{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ where $a_n = n^2 + n$, find the value of $a_3$. Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Orvin went to the store with just enough money to buy 30 balloons at the regular price. The store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 1/3 off the regular price. What is the greatest number of balloons Orvin could buy? Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If $(1 - 2x)^7 = a + a_1 x + a_2 x^2 + \ldots + a_7 x^7$, calculate the value of $a_1 + a_2 + \ldots + a_7$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = log2(x) in the interval [1, 2]. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $(x+5)(1-x)\geqslant 8$ and express the solution set in interval notation. | [-3, -1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Jun's travel box has a six-digit password, but he forgot the last digit. The last digit has 10 possibilities (0-9). What is the probability that Xiao Jun can open the travel box in one attempt? Express your answer as a fraction. | \frac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The complex number $2+i$ and the complex number $\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, respectively. Calculate the angle $\angle AOB$ in radians, using $\pi$ for pi. | \frac{\pi}{4} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$\overrightarrow {a}=(x-1, 2), \overrightarrow {b}=(4, y)$$, if $$\overrightarrow {a} \perp \overrightarrow {b}$$, find the minimum value of $$9^{x}+3^{y}$$. Provide your answer as a single number and explain when equality occurs. | 6 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan \alpha = \frac{1}{3}$, calculate the value of $\frac{\cos^2 \alpha - 2\sin^2 \alpha}{\cos^2 \alpha}$. Express your answer as a simplified fraction. | \frac{7}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Class 4 had a surplus of $82$ yuan in class funds at the end of the first semester. The income and expenses of the class funds for $4$ times are as follows (starting from $0$ yuan, income is positive): $+230$ yuan, $-75$ yuan, $-110$ yuan, and an unknown transaction. Calculate the amount of the last transaction that re... | 37 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_1 = 1/4 and a_3a_5 = 4(a_4-1), find the value of a_2. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of four-digit numbers that can be formed using the digits 0, 1, 2, 3 without repeating any digit, given that 0 cannot be the first digit. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A wheel with a fixed center and an outside diameter of 6 feet is required to cause a point on the rim to travel a distance of one mile. Calculate the number of revolutions of the wheel required to achieve this. Express your answer as a fraction with pi in the denominator. | \frac{880}{\pi} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors, $\overrightarrow{a}$ and $\overrightarrow{b}$, in a plane with an angle of $\frac{2\pi }{3}$ between them, where $\overrightarrow{a}=(3,0)$ and $|\overrightarrow{b}|=2$, calculate the value of $|\overrightarrow{a}+2\overrightarrow{b}|$. Express your answer in simplest radical form, if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation y = 2x + (3-a), find the range of values for a such that the line intersects the x-axis between points A(1,0) and B(3,0), inclusive of A and B. Express your answer as an inequality or interval. | [5,9] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the space rectangular coordinate system Oxyz, find the coordinates of the point symmetric to point A (-3, -4, 5) with respect to the plane xOz. Express your answer as a set of coordinates (x, y, z). | (-3, 4, 5) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a rhombus with side length 5, one of its diagonals is not greater than 6, and another is not less than 6, what is the maximum value of the sum of the lengths of the two diagonals? Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
There are two ponds at the bottom of Gabrielle's garden, each containing frogs and toads. In one pond, the ratio of frogs to toads is 3:4. In the other pond, the ratio of frogs to toads is 5:6. Suppose there are 36 frogs in total. What is the largest possible total number of toads in the ponds? Express your answer as a... | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of the binomial expression $(- \sqrt {x}+ \frac {1}{x})^{10}$. Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression: ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\ldots +{C}_{10}^{2}$. Express your answer as a single integer. | 165 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(5x+4)^{3}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$, calculate the value of $(a_{0}+a_{2})-(a_{1}+a_{3})$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of the rational number -2023. Express your answer as a single integer. | 2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
If the arithmetic mean of \(a\) and \(b\) is equal to twice their geometric mean, where \(a > b > 0\), calculate the ratio \(\frac{a}{b}\) and express your answer as the nearest integer. | 14 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A group of teachers and students totaling 328 people are going on a trip by bus. There is already a school bus that can accommodate 64 people. If they rent coaches, each of which can accommodate 44 people, set up an equation to represent the situation, where x is the number of coaches to be rented. Express your answer ... | 44x + 64 = 328 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given proposition p: If x > 1, then x > 2; proposition q: ∀x,y ∈ ℝ, if x+y ≠ 2, then x ≠ 1 or y ≠ 1, determine the true compound proposition using the logical operators ¬, ∧, and ∨. Express your answer as a single compound proposition (e.g., p ∧ q, ¬p ∨ q). | ¬p ∧ q | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{x|x^2-3x-4\leq0\}$, and the set $N=\{x|\ln(x)\geq0\}$, find the intersection $M\cap N$. Express your answer in interval notation. | [1, 4] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, one of its asymptotes is parallel to the line $2x - y + 1 = 0$. Find the eccentricity $e$ of the hyperbola. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Analyze the sign of the expression I, where I is defined as:
\[ I = (n+1)^2 + n - \left[\sqrt{(n+1)^2 + n + 1}\right]^2 \]
and n is a natural number. Express your answer as a single word or phrase describing the sign of I (e.g., "positive", "negative", "zero", etc.). | positive | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, there is a circle centered at the point (3, -5) with radius r. There are exactly two points on the circle that are at a distance of 1 from the x-axis. Find the range of values for the radius r. Express your answer as an inequality in the format a < r < b, where a and b are numerical ... | 4 < r < 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers z1 = 1 + i and z2 = 2 + bi, if the product of z1 and z2 is a real number, find the value of b. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a circle M that passes through the fixed point (0, 1) and whose center M moves along the parabola $x^2 = 2y$, calculate the length of the chord intercepted by the x-axis on circle M. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $(a+i)^2$ corresponds to a point on the negative half of the y-axis, calculate the value of the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ and $b$ be positive numbers such that $a + b = 2$. Calculate the minimum value of $\frac{1}{a+1} + \frac{4}{b+1}$. Express your answer as a single number in the format $\boxed{[answer]}$. | \frac{9}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. What is the minimum number of socks the Martian must remove from the drawer to be certain there will be 5 socks of the same co... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x+2)=2x+3$, find the expression for $f(x)$. Express your answer as a function of $x$, e.g. $f(x) =...$. | 2x-1 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If S_6 / S_3 = 3, find the value of S_12 / S_9. Express your answer as a numerical value. | \frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\cos 75^\circ \cos 15^\circ - \sin 75^\circ \sin 15^\circ$. Express your answer as a simplified numerical value. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the universal set $U$ is the set of real numbers $R$, and we have sets $M=\{x|x<1\}$ and $N=\{x|0<x<2\}$, find the intersection $M\cap N$. Express your answer as a set in the format $\{x|condition\}$ that satisfies both conditions of sets $M$ and $N$. | \{x|0<x<1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the height of a sheet of paper with a thickness of 0.1 millimeters after it has been folded in half 20 times, given that $2^{10} \approx 1000$. Express your answer in meters. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
If $z=1+2i$, then calculate the value of $\frac{4i}{z\overline{z}-1}$. Express your answer as a single value. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\left\{{x∈{N}|{\frac{1}{{x-2}}}≤0}\right\}$, determine the elements of set M. Express your answer as a set of natural numbers using standard set notation (e.g., {1, 2,...}). | \{1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Ali and Veli go hunting. The probability that each will successfully hit a duck is 1/2 on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. Calculate the probability that Veli hits more ducks than Ali. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $k$ in the hyperbola equation $5x^{2}-ky^{2}=5$ if one of its focal points is $(2,0)$. Express your answer as a single value. | \frac{5}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A school has 104 senior teachers, 46 intermediate teachers, and several other teachers. A stratified sampling of 42 teachers from all the teachers at the school was conducted for a survey, and it is known that 12 of the other teachers were included in the sample. Calculate the total number of teachers at the school. | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let $P=\{(x,y) | x+y=5, x\in\mathbb{N}^*, y\in\mathbb{N}^*\}$. Calculate the number of non-empty subsets of set P. Express your answer as a numerical value. | 15 | big_math | [
"Mathematics -> Combinatorics -> Counting Methods -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $x \in (-1,0)$, and define $\alpha_1 = \cos(\sin x\pi)$, $\alpha_2 = \sin(\cos x\pi)$, $\alpha_3 = \cos(x+1)\pi$. Determine the relationship among $\alpha_1$, $\alpha_2$, and $\alpha_3$. Express your answer in the format $\alpha_i < \alpha_j < \alpha_k$. | \alpha_3 < \alpha_2 < \alpha_1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through the point (1, 3) with a slope of 1. Express your answer in the standard form Ax + By + C = 0. | x-y+2=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $a+bi$ where $a, b \in \mathbb{R}$, what is the condition that must be met for its square to be a real number? Express your answer as an equation or condition. | ab = 0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In 2014, the Education Bureau of Ji'an City implemented a "support teaching" activity. A county-level middle school had 3 math teachers and 6 language teachers participating in "support teaching" activities in 3 township-level middle schools. Each township-level middle school was allocated 1 math teacher and 2 language... | 540 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
For the number 2020, find the count of its divisors that have more than 3 divisors themselves. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.0000037 in scientific notation. Provide your answer in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | $3.7 \times 10^{-6}$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
At a local farm, strawberries are sold at $16.50 for each basket. What is the total cost to buy 4 baskets of strawberries? Express your answer as a decimal number with two places, representing the total cost in dollars. | 66.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A circle is inscribed in a square of side m, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. Find the limit of the sum of the areas of the circles as the number of iterations grows beyond all bounds. Express your answer as a mathematical expression involving pi and... | \frac{\pi m^2}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression: \(1+1-2+3+5-8+13+21-34\). Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
If \( N = \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}} - \sqrt{3-2\sqrt{2}} \), calculate the value of \( N \). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-2x-b=0$ and knowing that one of its solutions is $x=-1$, calculate the other solution using Vieta's formulas. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given two points \( P(\cos \alpha, \sin \alpha) \) and \( Q(\cos \beta, \sin \beta) \) on the unit circle, find the maximum value of the distance between \( P \) and \( Q \). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence, the sum of the first n terms is denoted as S_n. If a_4 + a_9 = 10, find the sum of the first 12 terms, S_12. Express your answer as a single numerical value. | 60 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. What is the smallest possible total number of measurements (N) that satisfies the given percentages?
\[\begin{tabular}{c c}\text{measured value}&\text{percent frequency}\\ \hline 0 & 12.5\\ 1 & 0\\ 2 & 50\\ 3 & 25\... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the center of the circle $(x+1)^2 + (y-2)^2 = 4$ and has a slope of $1$. Express your answer in standard form (Ax + By + C = 0). | x - y + 3 = 0 | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The numbers 41, 35, 19, 9, 26, 45, 13, 28 are arranged in pairs so that the sum of the numbers in each pair is the same. Calculate the number that should be paired with 13 to achieve this equal sum. | 41 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A cube with an edge length of 2 has its vertices on the surface of a sphere. Calculate the volume of this sphere. Express your answer in terms of π. | 4\sqrt{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, if a_4 + a_6 = 12, and S_n is the sum of the first n terms of {a_n}, then find the value of S_9. Express your answer as a single numerical value. | 54 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A jar contains quarters (worth $0.25 each), nickels (worth $0.05 each), and pennies (worth $0.01 each). The value of the quarters is $10.00. The value of the nickels is $10.00. The value of the pennies is $10.00. If Judith randomly chooses one coin from the jar, what is the probability that it is a quarter? Express you... | \frac{1}{31} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If a_2 + a_8 = 10, find the sum of the first 9 terms, S_9. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle ${x^{2}}+{y^{2}}=9$, if the x-coordinates of all points on the circle remain unchanged and the y-coordinates become four times the original, find the equation of the resulting curve. Express your answer as an equation in the standard form of a circle or ellipse. | \dfrac{{x^{2}}}{9}+\dfrac{{y^{2}}}{144}=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations:
|x| + x + y = 10
x + |y| - y = 12
and find the exact value of x + y. Express your answer as a single fraction or integer. | \frac{18}{5} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let p and q be two consecutive terms of the sequence of odd primes. Find the minimum number of positive divisors of p + q. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A batch of 200 parts is sampled, with each part having a probability of 0.25 to be sampled. Calculate the sample size n. Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a > 0, b < 0, and |a| < |b|, determine the correct ordering of the variables a, b, -a, and -b based on their magnitude and sign. Express your answer using the greater than (>) symbol to denote the ordering. | -b > a > -a > b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Factor the quadratic expression $a^2 - 2a - 15$ using the cross-multiplication method. Express your answer as a product of two binomials. | (a + 3)(a - 5) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x) = x^2 + (m-1)x - 3$ is an even function, find the value of $m$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
For the function f(x) = x^3, find the number of tangent lines that have a slope equal to 1. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The Jiancheng Building has a total of 5 floors, each floor has two staircases. How many ways are there to go from the first floor to the fifth floor? Express your answer as a numerical value. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(1)=1$, calculate the limit $\lim_{\Delta x \to 0} \frac{f(1+3\Delta x) - f(1)}{\Delta x}$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality m ≤ 8x - 1 ≤ n, where the length of the interval of solutions is 25, find the value of n - m. Express your answer as a single integer. | 200 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2^{m} \cdot 2^{n} = 16$, find the value of $m+n$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $f(x)=\cos 2x+6\cos \left( \frac{\pi}{2}-x \right)$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Two coins are tossed. Given that one coin lands on heads, calculate the probability that the other coin also lands on heads. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality $(1-a)x^2 - 4x + 6 > 0$ is $\{x|-3 < x < 1\}$, and the solution set of $ax^2 + bx + 3 \geq 0$ is $\mathbb{R}$, find the range of values for $b$. Express your answer as an interval or a combination of intervals, including the endpoints if they are included. | [-6, 6] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $a_n$, if $a_2+a_4 = 6$, find the sum $a_1+a_2+a_3+a_4+a_5$. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \( N=\frac{1987^{4(\sqrt{3}+2)^{1987}+1988}}{1987^{(\sqrt{3}+2)^{1988}}+(\sqrt{3}+2)^{1988}} \), calculate the last digit of \( N \). Express your answer as a single digit (0-9). | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Select two numbers $a$, $b$ sequentially (without replacement) from the four numbers $1$, $2$, $3$, $4$. Calculate the probability that $a^{2} \geqslant 4b$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $x, y, z \in \mathbb{R}^*$, satisfying $x - 2y + 3z = 0$, find the minimum value of $$\frac {y^{2}}{xz}.$$ Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The relationship between the speed of sound v (in m/s) and the temperature t (in degrees Celsius) in the air is shown in the table below. If the speed of sound is 346 m/s, what is the temperature at that time? Express your answer in degrees Celsius as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(\frac{\pi}{6} - \alpha\right) = \cos\left(\frac{\pi}{6} + \alpha\right)$, find the value of $\cos 2\alpha$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = -x^3 + 3x on the interval [-2, 2]. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
50 students participated in the long jump and shot put tests, with 40 and 31 students passing the long jump and shot put tests, respectively. There were 4 students who failed both tests. Find the number of students who passed both tests. Express your answer as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$ with its right focus at $F_2(3,0)$, calculate the distance from the focus to its asymptote. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Consider the function f(x) = lg x + x - 3 and the interval (2,3). Given the bisection method results: f(2.25) < 0, f(2.75) > 0, f(2.5) < 0, and f(3) > 0, determine the interval in which the root of the equation lg x + x - 3 = 0 lies. Specify the interval in the form (a, b). | (2.5, 2.75) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} where each term is positive, and two terms a_m and a_n (m, n in N^{*}) such that sqrt{a_m*a_n} = 4a_1, and a_7 = a_6 + 2a_5, find the minimum value of 1/m + 4/n. Express your answer as a single fraction. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimal positive period of the function $f(x) = \frac{\tan x}{1 + \tan^2 x}$. Express your answer in terms of pi (π). | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the upper limit of the state space complexity of Go, $M \approx 3^{361}$, and the total number of atoms of ordinary matter in the observable universe, $N \approx 10^{80}$, calculate the value of $\dfrac{M}{N}$ using the logarithmic approximation $\lg 3 \approx 0.48$. Express your answer as a power of 10. | 10^{93} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of \(\frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\). Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $ax - y = 3$ and the solution set $\left\{\begin{array}{}x=1 \\ y=2\end{array}\right.$, find the value of $a$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1 = -1$, $a_2 = 2$, and $a_{n+1} = a_n + a_{n-1}$ (where $n \in \mathbb{N}^*$). Calculate $a_{2008}$. Provide your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence \( \{a_n\} \) whose sum of the first \(n\) terms is \(S_n = 5n^2 + kn - 19\) (\(n \in \mathbf{N}^*\)), and \(a_{10} = 100\), find the value of \(k\). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
An equilateral triangle has sides of \(2x\) and \(x+15\). Calculate the perimeter of the triangle. Express your answer as a whole number. | 90 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
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