instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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Given that f(x) and g(x) are odd functions, and F(x) = a f(x) + b g(x) + 2 has a maximum value of 8 in the interval (0, +∞), find the minimum value of F(x) in the interval (-∞, 0). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the contrapositive of the proposition "If x and y are both even, then x+y is also even". Express your answer as a logical statement in the format "If..., then...". | If x+y is not even, then x and y are not both even. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A cuboid has three faces meeting at one vertex with areas of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{6}$, respectively. Calculate the length of the diagonal of this cuboid. Express your answer as a simplified radical. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequalities: \( r < s \), \( t > q \), \( q > p \), and \( t < r \), determine the greatest number among \( p, q, r, s, t \). Express your answer as a single variable. | s | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the coordinates of the vertices of triangle $\triangle ABC$ as $A(2,-1)$, $B(3,2)$, $C(-5,4)$, find the length of the median $AD$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_n\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \cdot a_{24} < 0$, find the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$. Express your answer as a positive integer. | 46 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that an ellipse is defined by the equation $4x^{2}+ky^{2}=4$ and one of its foci is located at the coordinate $(0,1)$, find the value of the real number $k$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the universal set is the set of real numbers, set A is defined as the closed interval from -1 to 4, and set B is defined as the closed interval from 3 to 5, find the intersection of A and the complement of B. Express your answer in interval notation. | [-1,3) | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that α is an angle in the fourth quadrant and $\tan\alpha=-\frac{5}{12}$, calculate the value of $\sin\alpha$. Express your answer as a fraction, including the negative sign if applicable. | -\frac{5}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Two ships, one 200 meters in length and the other 100 meters in length, travel at constant but different speeds. When traveling in opposite directions, it takes 10 seconds for them to completely pass each other. When traveling in the same direction, it takes 25 seconds for them to completely pass each other. Calculate ... | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets:
\[ A = \{-1, 0, 2\} \]
\[ B = \{-x \mid x \in A \text{ and } 2-x \notin A\} \]
Determine the elements of set \( B \). Express your answer as a set of integers using standard set notation. | \{1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $i$ is the imaginary unit. Find the value of $a$ such that the complex number $a - \\frac{10}{3-i}$ is a pure imaginary number. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the piecewise function $$g(x)= \begin{cases} \cos \frac {\pi}{2}x, & x\leq0 \\ \log_{4}(x+1), & x>0\end{cases}$$ and find the number of pairs of points symmetric about the origin that lie on the graph of g(x). Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{1}{x} - \frac{1}{y} = \frac{1}{z}$, calculate the value of $z$. Express your answer as a fraction in terms of $x$ and $y$. | \frac{xy}{y-x} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A, B, C, D, and E are five people working on a project. If A, B, C, and D work together, it takes 6 days to complete the project; if B, C, D, and E work together, it takes 8 days to complete the project; if A and E work together, it takes 12 days to complete the project. Calculate the number of days E needs to complete... | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average of two numbers is 12 and the average of three other numbers is 7, find the average of all five numbers. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence {a_n}, if log a_4 + log a_7 + log a_{10} = 3, then find the value of a_1a_{13}. Express your answer as a numerical value. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Jones's four cousins have ages that are four different single-digit positive integers. The product of the ages of two of them is 24, and the product of the ages of the other two is 30. What is the sum of the ages of the four cousins? Express your answer as a single integer. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $2x^{2}-x-5=0$, identify the coefficients of the quadratic term and the linear term, in that order, separated by a comma. | 2, -1 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Using Euler's formula $e^{ix} = \cos x + i\sin x$, find the product of the complex numbers $z_{1} = e^{i\frac{\pi}{3}}$ and $z_{2} = e^{i\frac{\pi}{6}}$. Express your answer as a complex number in the form a + bi. | i | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
We are preparing to distribute 7 identical fitness equipment to 5 different communities, where communities A and B each must receive at least 2 pieces of equipment, and the other communities may receive none. Calculate the total number of different distribution schemes. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\sqrt{3x} \cdot \sqrt{\frac{1}{3}xy}$. Express your answer as a simplified mathematical expression. | x\sqrt{y} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the monomials $-3a^{m}b^{2}$ and $\frac{1}{2}{a^2}{b^n}$ is still a monomial, find the value of $mn$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta$, $|\overrightarrow{b}|=2|\overrightarrow{a}|$, and $\overrightarrow{a}•\overrightarrow{b}={\overrightarrow{a}}^{2}$, find the value of $\theta$ in radians. If your answer involves $\pi$, express it as a fraction ... | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line with a slope angle of 120° and a y-intercept of -2. Express your answer in slope-intercept form (y = mx + b), including all necessary mathematical symbols and notation. | y = -\sqrt{3}x - 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, square ABCD has a side length of 4 units. Find the total area of the shaded regions. Express your answer in square units as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In a regular polygon, each interior angle measures 120°. Calculate the number of diagonals that can be drawn from a vertex of this polygon. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $(\log_{2}3)^x - (\log_{5}3)^x \geq (\log_{2}3)^{-y} - (\log_{5}3)^{-y}$, derive the condition on x and y. Express your answer as an inequality involving x and y. | x+y \geq 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A 100 yuan bill is to be exchanged for 20 yuan bills and 10 yuan bills. Calculate the total number of ways this can be done. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $z(1-i)=2-3i$, calculate the modulus of $z$, denoted as $|z|$. Express your answer as a simplified radical. | \frac{\sqrt{26}}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Several people plan to pool their money to buy an item. If each person contributes $12, they will be $3 short; if each person contributes $13, they will have $12 left over. Find the total number of people. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $4+(-8)\div (-4)-(-1)$. Provide your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. What is the smallest possible number of guesses, n, that Berk needs to guarantee finding Ayca's number? Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be a function defined on R with derivative f′(x). If f(x) + f′(x) < 1 and f(0) = 2016, find the solution set of the inequality e^x*f(x) - e^x > 2015. Express your answer as an interval or a combination of intervals. | (-\infty, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that positive real numbers a and b satisfy a + 2b = 1, find the value of a when the product ab is maximized. | \frac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 3xf'(1), find the value of f'(2). | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_{n}} where all terms are positive numbers, if a_{1}a_{3} = 36 and a_{4} = 54, then find the common ratio q. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,3)$, find the magnitude of $|2\overrightarrow{a} - \overrightarrow{b}|$. Express your answer as a radical if necessary. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola y = 4ax^2 (a ≠ 0). Express your answer as an ordered pair (x, y) in terms of the variable 'a'. | (0, \frac{1}{16a}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a cinema, the first row has $m$ seats, and each subsequent row has one more seat than the row in front of it. Find the number of seats in the $n$-th row. Express your answer as an algebraic expression in terms of $m$ and $n$. | m + n - 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The northern village has 8,758 people, the western village has 7,236 people, and the southern village has 8,356 people. A total of 378 people are to be conscripted from the three villages according to their population size. How many people should be conscripted from the western village? Express your answer as a whole n... | 112 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 4 male students and 2 female students, 3 individuals are randomly selected to participate in a speech competition. Let the random variable ξ represent the number of female students among the selected 3 individuals. Calculate P(ξ ≤ 1), the probability of selecting at most 1 female student. Express your a... | \dfrac{4}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The war between Russia and Ukraine has been ongoing since February 24th of last year, mainly due to the continuous instigation by NATO led by the United States. According to incomplete statistics, the United States alone has provided military aid worth $27.58$ billion to Ukraine. Express $27.58$ billion in scientific n... | 2.758\times 10^{10} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that ABCD is a parallelogram and AE = 2EB, express EC as a linear combination of AB and AD, and calculate the sum of the coefficients lambda and mu. Express your answer as a single fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let x = cos 36° - cos 72°. Using double angle formulas for cosine, express cos 36° and cos 72° in terms of y = cos 36° and simplify the resulting expression to find the exact value of x. Express your answer as a single fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $1101_{(2)}$ to a base-5 number. Express your answer in base-5 notation, using subscript (5) to denote the base. | 23_{(5)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A proposed new $5 coin is called the "foonie". The foonie's two faces are identical and each has an area of 5 cm^2. The thickness of the foonie is 0.5 cm. Calculate the number of foonies in a stack that has a volume of 50 cm^3. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The temperature drops by 5℃ from -6℃. Calculate the new temperature in degrees Celsius (℃). | -11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$, where $a_4 + a_8 = 16$, calculate the sum of the first 11 terms of the sequence, denoted $S_{11}$. Express your answer as a single numerical value. | 88 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the units digit in the product \((5+1)(5^3+1)(5^6+1)(5^{12}+1)\). Express your answer as a single digit. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In a box, there are 3 red balls and 2 white balls, all of the same shape and size. If two balls are drawn successively without replacement, and the first ball drawn is white, calculate the probability of drawing a red ball second. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2^x + 2^{-x}, if f(a) = 3, then find the value of f(2a). Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)$, given that $xy \not= 0$. Express your answer as a simplified algebraic expression. | 2xy+\frac{2}{xy} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The temperature of the refrigeration compartment of a household refrigerator is 4°C, and the temperature of the freezer compartment is 22°C lower than that of the refrigeration compartment. Find the temperature of the freezer compartment. Express your answer in degrees Celsius (°C) as an integer, including a negative s... | -18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-2x^{2}y)^{3}$. Express your answer as a simplified algebraic expression. | -8x^{6}y^{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Polynomial Operations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the integral $$\int_{2}^{4} \dfrac{1}{x} \, dx.$$ Express your answer as a single mathematical expression. | \ln(2) | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\left\{\begin{array}{l}{x=3}\\{y=2}\end{array}\right.$ is a solution to the two-variable linear equation $ax+2y=1$, calculate the value of $a$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, the equation $a_1 + 3a_8 + a_{15} = 120$ holds. Find the value of $a_8$. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \alpha = 3$, simplify the expression $\frac {4\sin \alpha - 2\cos \alpha}{5\cos \alpha + 3\sin \alpha}$. Express your answer as a simplified fraction. | \frac{5}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arithmetic sequence ${a_n}$ has a common difference of 3, and $a_1 + a_3 = 8$, find the value of the sum $S$ of the first 4 terms of the sequence ${a_n}$. Express your answer as a single numerical value. | 22 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {1,2,3,4,5}, set A = {1,3,4}, and set B = {2,4}, find the result of the set operation (∁_U A) ∪ B. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas. | {2,4,5} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \(\{a_n\}\), where \(a_1 = 4\) and \(d = 2\), calculate the value of \(a_3\) using the formula \(a_n = a_1 + (n-1)d\). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (x-2)(x^2+c) with an extremum at x=2, find the slope of the tangent line to the graph of f(x) at x=1. Express your answer as a single numerical value. | -5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2\cos ( \frac{π}{2}-α)-\sin ( \frac{3}{2}π+α)=- \sqrt{5} $, find the value of $ an α$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A sequence $\{a_n\}$ consists of 5 terms, where $a_1 = 0$, $a_5 = 2$, and $|a_{i+1} - a_i| = 1$ for $i=1, 2, 3, 4$. Calculate the number of different sequences that satisfy these conditions. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose {1/(a_n+1)} is an arithmetic sequence, a_3 = 2, a_7 = 1. Calculate the value of a_11. Express your answer as a single value in the form \boxed{a_11}. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function $f(x)$ is $f′(x)$, and they satisfy the relationship $f(x)=x^{2}+3xf′(2)$, find the value of $f′(2)$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is a function defined on the real numbers, and for any real number x, it satisfies f(x+4) = -f(x) + 2√2. If the graph of the function y = f(x-1) is symmetric about the line x=1, calculate the value of f(2014). Express your answer as a single value, potentially involving square roots. | √2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Two men and two women are lined up in a row. Find the total number of ways they can be arranged alternately by gender. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A reporter wants to take a photo of 4 volunteers and the elder they are helping. They need to be arranged in a row, with the elder required to be in the middle. Calculate the number of different arrangements possible. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.-2 \lt x \lt 2\}$ and $B=\{x|x＞\sqrt{3}\}$, find the intersection of sets A and B, denoted as A ⋂ B. Express your answer in interval notation. | (\sqrt{3}, 2) | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of -2022. The opposite of a number is defined as the number that, when added to the original number, results in zero. Express your answer as a single integer. | 2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that one root of the equation $ax^{2}+bx+c=5$ with respect to $x$ is $2$, and the axis of symmetry of the quadratic function $y=ax^{2}+bx+c$ is the line $x=2$, find the coordinates of the vertex of this parabola. Express your answer as a coordinate pair (x, y). | (2,5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial (x-m)(x-n), what is the condition on m and n that makes the linear term in x zero? Express your answer as an equation involving m and n. | m + n = 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomial Operations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (\cos5°, \sin5°)$ and $\overrightarrow{b} = (\cos65°, \sin65°)$, calculate the magnitude of $\overrightarrow{a} + 2\overrightarrow{b}$. Express your answer as a single value. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets M={y | x+y=2} and N={(x,y) | x-y=4}, determine the intersection of sets M and N. Express your answer as an ordered pair (x,y). | (3,-1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the "optimized sum" of the sequence $a_1, a_2,..., a_{2018}$ is 2019, calculate the "optimized sum" of the sequence $2, a_1, a_2,..., a_{2018}$. Express your answer as a single number. | 2020 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Nine cards numbered from 1 to 9 were placed face down on the table. Aleksa, Bart, Clara, and Deindra each picked up two of the cards.
- Aleksa said, "My numbers add up to 6."
- Bart said, "The difference between my numbers is 5."
- Clara said, "The product of my numbers is 18."
- Deindra said, "One of my numbers is tw... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If $3^{x}=15$ and $3^{y}=5$, then find the value of $3^{x-y}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. Calculate the perimeter of the remaining quadrilateral. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable $X$ follows a binomial distribution $X\sim B(4,\frac{1}{2})$, find the probability $P(X=2)$. Express your answer as a simplified fraction. | \frac{3}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the product of two hundred thousand times two hundred thousand. Express your answer as a single number in standard form. | 40,000,000,000 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A factory distributes 11410 kg of rice equally into 3260 bags. A family uses 0.25 kg of rice each day. How many days would it take this family to use up one bag of rice? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $x^2+y^2+4x-2y-4=0$, find the coordinates of the center and the length of the radius. Express your answer in the format $(x, y)$ for the center coordinates and a numerical value for the radius length. | (-2, 1), 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select a number x in the interval [0, π]. What is the probability of the event "cos(x) > 1/2"? Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
For the inverse proportion function y = (m-5)/x, where y increases as x increases on each branch of its graph, determine the range of values for m. Express your answer as an inequality. | m < 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If the square roots $\sqrt{5b}$ and $\sqrt{3+2b}$ can be combined, find the value of $-b$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The rectangle $\mathrm{ABCD}$ represents a piece of land in the shape of a rectangle whose width is $\frac{3}{5}$ of its length. The rectangle ABEF represents a garden in the shape of a rectangle whose width is also $\frac{3}{5}$ of its length. Find the ratio of the area of the garden to the total area of the land. Exp... | 36\% | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Randomly split 2.5 into the sum of two non-negative numbers. Then, round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3? Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are positive numbers and satisfy the equation 2x + y - 3 = 0, find the minimum value of 2/x + 1/y. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
How many x are there such that x, [x], {x} are in harmonic progression, where [x] is the largest integer less than or equal to x and {x} = x - [x]? Express your answer as a non-negative integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ζ follows a normal distribution N(4, 6^2), and P(ζ ≤ 5) = 0.89, determine P(ζ ≤ 3). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(k,4)$, and $\overrightarrow{a}\parallel \overrightarrow{b}$, calculate the value of the real number $k$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=2+i$, calculate the value of $\frac{\overline{z}}{z}$. Express your answer as a complex number in the form $a + bi$. | \frac{3}{5}-\frac{4}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A polynomial subtracted by $x^{2}-3y^{2}$ equals $x^{2}+2y^{2}$. Find the polynomial. Express your answer as a polynomial in terms of $x$ and $y$. | 2x^{2} - y^{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y= \sqrt{3}\tan \frac{x}{2}$, with $x\neq (2k+1)\pi (k\in \mathbb{Z})$, find the expression for $x$ when $y=1$. Express your answer as an equation for $x$ in terms of $k$ and $\pi$. | 2k\pi + \frac{\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Consider the plane region represented by the system of inequalities \(\begin{cases} & y \leqslant x, \\ & 3y \geqslant x, \\ & x+y \leqslant 4 \end{cases}\\). Let this region be denoted as $\Omega_1$. Additionally, the plane region represented by the inequality $${(x+2)^{2}}+(y-2)^{2}\leqslant 2$$ is denoted as $\Omega... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vertex of angle α is at the origin, and its initial side coincides with the non-negative half-axis of the x-axis. If point P(-2,-1) is a point on the terminal side of angle α, calculate 2cos²α - sin(π - 2α). Express your answer as a fraction or decimal. | \frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, the first term $a_1=1$ and the common ratio $q=2$. Calculate the value of the fifth term $a_5$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function $y=-2\left(x+3\right)^{2}-5$, identify the coordinates of its vertex. Express your answer as an ordered pair $(h,k)$. | (-3,-5) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
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