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 |
|---|---|---|---|---|---|
A right triangle has one leg with a length of 11, and the lengths of the other two sides are also natural numbers. Calculate the perimeter of the triangle. Express your answer as a single integer. | 132 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function y = -cos^2x + 2sin x + 2. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \\begin{cases} \\log_{9}x, x > 0 \\\\ 4^{-x} + \\frac{3}{2}, x \\leqslant 0 \\end{cases}$, calculate the value of $f(27) + f(-\\log_{4}3)$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let $y=f(x)$ be an odd function defined on $\mathbb{R}$, and $f(x)$ is decreasing on the interval $(0,+\infty)$, with $f(2)=0$. Find the solution set for $f(x) > 0$. Express your answer as an interval or a union of intervals. | (-\infty,-2)\cup(0,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the modulus of the complex number (a + 2i)(1 + i) is 4, find the value of the real number a. Express your answer as a real number, including the ± symbol if applicable. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z satisfies iz = 1 - 2i, calculate the magnitude of z. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $-1-3$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The terms of an arithmetic sequence $\{a_{n}\}$ are all positive. If $a_{3}+2a_{_{6}}=6$, what is the maximum value of $a_{4}a_{_{6}}$? Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}| = 1$ and form a $\frac{\pi}{2}$ angle between them, calculate the value of $\overrightarrow {a} \cdot (-6\overrightarrow {a} - \overrightarrow {b})$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a>0$ and $b>0$, find the maximum value of $m$ that satisfies the inequality $$\frac {4}{a}+ \frac {1}{b} \geq \frac {m}{a+4b}.$$ | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(a, b, c) = (c+a)/(c-b), evaluate f(1, -2, -3). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an even function defined on R and monotonically increasing on the interval [0, +∞), derive the range of values for the real number 'a' that satisfies the inequality f(a*2^x) - f(4^x + 1) ≤ 0. Express your answer as an interval. | [-2, 2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $$a+ \frac {1}{a}=7$$, calculate the value of $$a^{ \frac {1}{2}}+a^{- \frac {1}{2}}$$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A cylindrical water tower with a base radius of 1 meter is being filled with a hose. After filling for 8 hours, the water level is 3 meters below the top of the tower. After filling for another 2 hours, it is still 2π cubic meters away from being full. Calculate the height of the cylindrical water tower. Express your a... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Use the method of successive subtraction to find the greatest common divisor (GCD) of 459 and 357. How many times do you need to perform subtraction to reach the GCD? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the plane rectangular coordinate system xOy, find the point symmetric to point (2,5) with respect to the y-axis. Express your answer as an ordered pair (x, y). | (-2,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Among 100 students in a school's reading club, 90 students have read either *Journey to the West* or *Dream of the Red Chamber*, 80 students have read *Dream of the Red Chamber*, and 60 students have read both *Journey to the West* and *Dream of the Red Chamber*. Calculate the total number of students who have read *Jo... | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^3$ term in the expansion of $(\\sqrt{x} - 2x)^5$. Express your answer as a single integer. | -10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function expression y = x^2 + 1 and the range {1,3}, how many homologous functions can be defined? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the approximation \( 1 - y \) for the value of \( \frac{1}{1 + y} \) where \( |y| < 1 \), find the ratio of the error made to the correct value. Express your answer as a mathematical expression. | y^2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Among 5 products, there are 2 defective ones, and the rest are qualified. Now, if we randomly select 2 out of these 5 products, what is the probability of exactly one being defective? Express your answer as a decimal value. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
What is the last digit of the number \( 2^2 + 20^{20} + 200^{200} + 2006^{2006} \)? Provide your answer as a single digit. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the regression equation $$\hat y = 4.75x + 257$$, calculate the estimated value of y when x = 28. Express your answer as a single numerical value. | 390 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 1 - e^x, find the equation of the tangent line to the curve at the point where it intersects the x-axis (point P). Express your answer in the form y = mx + b, where m is the slope and b is the y-intercept. | y = -x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A square has a perimeter of 24 units. A rectangle has the same area as the square. If the width of the rectangle is 4 units, what is the perimeter of the rectangle? Express your answer in units as a whole number. | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$, with at least one zero in the interval $(0, 4)$ and $f(0) = 0$, determine the minimum number of zeros the function $y = f(x)$ has in the interval $(-8, 10]$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $i^{2n-3}+i^{2n-1}+i^{2n+1}+i^{2n+3}$ for any integer n. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n}, where a_1 = 1, and a_n = n(a_{n-1}) for n in N*, find the general term of the sequence {a_n}. Express your answer as a mathematical expression in terms of n, using standard mathematical notation. | a_n = n! | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Throw a coin twice. Let event A be "the first time is heads" and event B be "the second time is heads". Calculate the conditional probability P(B|A). Express your answer as a fraction or decimal value between 0 and 1. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The largest desert in the world, the Sahara Desert in Africa, has a length of approximately $4800000m$. Convert this number into scientific notation, expressing your answer in the form $a \times 10^{b}$. | 4.8 \times 10^{6} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors a_m = (m, 1) and b_n = (2, n), where m, n ∈ {1, 2, 3}, denote the event'making a_m ⊥ (a_m - b_n) hold' as event A. Calculate the probability of event A occurring. Express your answer as a fraction in lowest terms. | \frac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, a smaller square lies inside a larger square. The perimeter of the smaller square is 72 cm. The shaded area is 160 cm². Find the perimeter of the larger square. Express your answer in centimeters as a whole number. | 88 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z(1+i)=3+2i$, find the imaginary part of the complex number $z$. Express your answer as a single fraction or decimal. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A regular triangular pyramid with all lateral faces being right triangles has a base edge length of 2. Calculate the total surface area of this pyramid. Express your answer as a simplified mathematical expression. | \sqrt{3} + 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle has a length that is 3cm longer than its width. If the area of the rectangle is 4cm^2, what is the length of the rectangle? Express your answer in centimeters (cm) as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vector $\overrightarrow{a} = (2, 4)$ is collinear with the vector $\overrightarrow{b} = (x, 6)$, determine the value of the real number $x$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Using three different weights of 1 gram, 3 grams, and 9 grams, various weights of objects can be measured. Assuming the objects to be measured and the known weights can be placed on either side of the balance scale, how many different weights of objects can be measured? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function y = (sin x + cos x)(sin x - cos x). Express your answer in radians. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A line $2x - y - 1 = 0$ is intersected by a circle $(x - 2)^2 + (y + 2)^2 = 9$. Determine the length of the chord formed by their intersection. Express your answer as a numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A point moves along a straight line. If the distance traveled s after t seconds from the starting point is given by the equation s = (1/3)t^3 - (1/2)t^2 - 2t + 1, find the time when the velocity is zero. Express your answer in seconds. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-1, 0) and B(3, 2), calculate the vector $\frac{1}{2} \overrightarrow{AB}$. Express your answer as an ordered pair (x, y). | (2, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A wooden stick is sawed into 5 segments, which took a total of 20 minutes. Calculate the average time it takes to saw once. Express your answer in minutes. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y=f(x) is defined in the vicinity of x0 and satisfies f(x0+2Δx)-f(x0)=2Δx+(Δx)2, find the derivative f'(x0) using the limit definition of a derivative. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Let
\[ N = 69^{5} + 5 \cdot 69^{4} + 10 \cdot 69^{3} + 10 \cdot 69^{2} + 5 \cdot 69 + 1. \]
Calculate the number of positive divisors of \( N \). Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
What is the probability of selecting the letter "n" from the word "polynomial"? Express your answer as a fraction. | \frac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) = a^x, g(x) = x^(1/3), h(x) = log_a x, where the real number a satisfies log_a(1 - a^2) > 0. Determine the correct ordering of the functions for x > 1. Express your answer in the form "h(x) < f(x) < g(x)" or a similar ordering. | h(x) < f(x) < g(x) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a box containing 3 screw base bulbs and 7 bayonet base bulbs, all of which have the same shape and power and are placed with their bases down. An electrician needs a bayonet base bulb and takes one bulb at a time without returning it. Calculate the probability that he only gets a bayonet base bulb on his third tr... | \frac{7}{120} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, $i$ and $j$ are unit vectors in the same direction as the positive x-axis and y-axis, respectively. Three points A, B, and C in the plane satisfy $\overrightarrow{AB} = 4\overrightarrow{i} + 3\overrightarrow{j}$, $\overrightarrow{AC} = k\overrightarrow{i} - \frac{1}{2}\overrightarrow... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). Find the ratio $PA:AB$. Express your answer as a ratio of two integers. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an equilateral triangle ABE inside square ABCD, where point F is the intersection of diagonal BD and line segment AE, and AB has length sqrt(1 + sqrt(3)), calculate the area of triangle ABF. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In an unusual country, there are three kinds of coins: Exes, Wyes, and Zeds. The value of 2 Exes equals the value of 29 Wyes, and the value of 1 Zed equals the value of 16 Exes. Find the value of 1 Zed in terms of Wyes. Express your answer as a single number. | 232 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y=f(x+1)$ is $[-1,3]$, determine the domain of $y=f(x^{2})$. Express your answer as an interval. | [-2,2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( x = 0.1234567891011 \cdots 998999 \), a decimal formed by concatenating the numbers starting from 1 to 999. Find the digit at the 1983rd position after the decimal point. Express your answer as a single digit. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of the number 2024. Express your answer as an integer. | -2024 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \u03c9 is a cube root of unity, simplify the expression (1-\u03c9)(1-\u03c9^2)(1-\u03c9^4)(1-\u03c9^8) and express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Abstract Algebra -> Field Theory"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If $\sqrt{63n}$ is an integer, find the smallest value of the positive integer $n$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality 2x - 3 > 7 - x for x. Express your answer as an inequality. | x > \frac{10}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x|0 < x < 4\}$ and $N=\{x|\frac{1}{3}\leqslant x\leqslant 5\}$. Find the intersection $M \cap N$ and express your answer as a set definition in the format $\{x|\text{condition}\}$. | \{x|\frac{1}{3}\leqslant x < 4\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=a^x+1$, where $a>0$ and $a \neq 1$, determine the point that the graph of this function must pass through. Express your answer as an ordered pair $(x, y)$. | (0, 2) | big_math | [
"Mathematics -> Algebra -> Precalculus -> Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
John scored 93 points in the current AHSME. If the old scoring method was still in effect, he would score only 84 points with the same answers. The new scoring method awards 5 points for each correct answer, 0 points for each incorrect answer, and 2 points for each unanswered question. The old scoring method starts wit... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arithmetic mean of two positive numbers a and b is 7/2, and the geometric mean is 2*sqrt(3), with a < b, find the eccentricity e of the hyperbola x^2/a^2 - y^2/b^2 = 1. Express your answer as a fraction or decimal value. | \frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If the simplest quadratic radical \\sqrt{a+2} is of the same type as the simplest quadratic radical \\sqrt{3a}, find the value of a that satisfies this condition. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, $a_2 + a_6 = 10$. Find the value of $a_1 + a_2 +... + a_7$. Express your answer as a single numerical value. | 35 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all prime numbers within 10. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 balls of the same size in a bag, including 2 red balls and 3 white balls. Balls are drawn from the bag without replacement. If it is known that a white ball is drawn on the first draw, calculate the probability of drawing a red ball on the second draw. Express your answer as a decimal value. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 3 - 2i, what is the imaginary part? Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a_n$ ($n=2,3,4,\ldots$) be the coefficient of the term containing $x$ in the expansion of $(3- \sqrt{x})^n$. Calculate the value of $$\frac{1}{a_2} + \frac{3}{a_3} + \frac{9}{a_4} + \ldots + \frac{3^{2011}}{a_{2013}}$$ and express your answer as a single fraction. | \frac{4024}{2013} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a box containing 3 screw base bulbs and 7 bayonet base bulbs, all of which have the same shape and power and are placed with their bases down, an electrician needs a bayonet base bulb and takes one bulb at a time without returning it. Calculate the probability that he only gets a bayonet base bulb on his third tr... | \frac{7}{120} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If the monthly "progress" rate and "regress" rate are both 20%, calculate the number of months it will take for the "progress" to be ten thousand times the "regress". Express your answer as a whole number. | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of ways to assign 4 volunteers to 3 different schools for surveying extracurricular activity content, ensuring that each school is allocated at least one volunteer. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the derivative of the function f(x) as f'(x)=x^{2}-4x+3, find the monotonically decreasing interval of f(x+1). Express your answer as an interval (a, b). | (0, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} ... | 1.2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the standard equation of ellipse $C$ is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b>0$), where the left and right foci are $F_{1}$ and $F_{2}$ respectively, and $P$ is the top point of the ellipse. If $\angle F_{1}PF_{2}=90^{\circ}$, calculate the eccentricity of the ellipse. Express your answer as a simplified fra... | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, $a_3 + a_9 = 4$. Find the sum of the first 11 terms, $S_{11}$, of the sequence ${a_n}$. Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Five students are standing in a row to take a photo. If two students, A and B, must stand next to each other, calculate the total number of different possible arrangements. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(0,σ^2), and P(ξ > 2) = 0.023, calculate the probability P(-2 ≤ ξ ≤ 2). Express your answer as a decimal value between 0 and 1. | 0.954 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Jia, Yi, Bing, Ding, and Wu participated in a 100-meter race. After the race, Ding said, "I ran faster than Yi." Bing said, "Wu crossed the finish line ahead of me." Jia said, "I ranked ahead of Ding and behind Bing." Based on their statements, determine the order of their rankings, expressing your answer as a sequence... | Wu > Bing > Jia > Ding > Yi | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangle ABC, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. If $a+b=21$ and $c=15$, then find the area of triangle ABC. Express your answer as a whole number. | 54 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Use the completing the square method to solve the quadratic equation $x^{2}-4x-2=0$. Express your answer in the form $(x-a)^{2}=b$, where $a$ and $b$ are the values obtained after completing the square. | $(x-2)^{2}=6$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Zara has collected 4 marbles, each made of agate, bumblebee stone, steel, and tiger-eye stone. She wants to arrange these 4 marbles in a single row on a shelf, but the steel and tiger-eye stone marbles must not be adjacent. Calculate the number of different arrangements possible. Express your answer as a single integer... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2cos(ωx + φ) and the condition that for any x ∈ ℝ, f(π/3 + x) = f(π/3 - x), define the function g(x) = 3sin(ωx + φ) - 1. Calculate the exact value of g(π/3). | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( a \) and \( b \) are non-zero real numbers, there are three distinct points \( A \), \( B \), and \( C \) on the parabola \( y=ax^2 + b \) such that \( BC \) is parallel to the x-axis and \(\angle BAC = 90^\circ \). Find the distance from point \( A \) to the line \( BC \). Express your answer as a fracti... | \frac{1}{|a|} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of the first three terms is 9 (a_1+a_2+a_3=9) and the sum of the next three terms is 27 (a_4+a_5+a_6=27). Find the sum of the next three terms (a_7+a_8+a_9). Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Two boys and three girls stand in a row for a photo. If among the three girls, exactly two girls stand next to each other, then how many different arrangements are possible? Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x|x|, find the value of x_0 for which f(x_0) = 4. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, the points representing the numbers $-1$ and $2023$ are denoted as point $A$ and point $B$, respectively. Find the distance between point $A$ and point $B$. Express your answer as a positive integer. | 2024 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(0,1)$, calculate the coordinates of $\overrightarrow{a}-2\overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (-1,0) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The famous mathematician of the Southern Song Dynasty, Qin Jiushao (circa 1202-1261), discovered a method for calculating the area of a triangle, known as the 'three obliques method.' The book 'Mathematical Treatise in Nine Sections' records his method as follows: 'Take the square of the smaller oblique side and add th... | 6 \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the intersection point of line $l_1: x + y - 2\sqrt{2} = 0$ and line $l_2: \begin{cases} x= \frac{\sqrt{2}}{2}t \\ y= \frac{\sqrt{2}}{2}t \end{cases}$ to the origin $O$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system xOy, find the area of the region enclosed by the curve 2|x| + 3|y| = 5. Express your answer as a simplified fraction or decimal. | \frac{25}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a, b, c, d can only be -1, 1, or 2, calculate the maximum possible value of ad - bc. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of solutions to the equation sin(x^sin(x)) = cos(x^cos(x)) in the closed interval [π/4, π/2]. Express your answer as a whole number. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1, 0, 1\}$ and $N=\{y|y=1+\sin\left(\frac{\pi x}{2}\right), x\in M\}$, find the number of proper subsets of the set $M\cap N$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
By placing addition signs between some of the digits of the number 123456789, we can obtain various sums. What is the minimum number of addition signs necessary to obtain the number 54 in this manner? Provide your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Compute the value of $\frac{\sin {110}^{\circ } \sin {20}^{\circ }}{{{\cos }^{2}}{155}^{\circ }-{{\sin }^{2}}{155}^{\circ }}$. Express your answer as a single fraction or decimal value. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Sarah is leading a class of 35 students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands 3 times, what is the greatest possible number of students that could have been standing before her third wa... | 31 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Cicely had her 21st birthday in 1939. In what year did she have her 100th birthday? Express your answer as a year in the format YYYY. | 2018 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin \alpha = \frac{4}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $\tan \alpha$. Express your answer as a fraction or decimal. | -\frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \alpha = 2$, calculate the value of $\sin \alpha \cos \alpha$. Express your answer as a fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Climbing a staircase with four steps, calculate the total number of possible ways to do so, considering different step sequences as distinct arrangements. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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