instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are not parallel, and vector $λ \overrightarrow{a} + \overrightarrow{b}$ is parallel to $\overrightarrow{a} + 2 \overrightarrow{b}$, determine the value of the real number $λ$. Express your answer as a single real number. | \dfrac {1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=i(3-2i)$, where $i$ is the imaginary unit, calculate $\overset{.}{z}$. Express your answer as a complex number in the form $a+bi$, including the imaginary unit $i$. | 2-3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a light ray is emitted from point M(2,1), reflects off the x-axis, and then reaches point N(4,5), calculate the total distance the light ray travels. Express your answer in simplest radical form. | 2\sqrt{10} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The function \( f(x) \) satisfies \( f(2 + x) = f(2 - x) \) for all real numbers \( x \). If the equation \( f(x) = 0 \) has exactly four distinct real roots, find the sum of these roots. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A random variable $\xi$ has a distribution with $P\left(\xi =k\right)=\dfrac{c}{k\left(k+1\right)}$, where $k=1$, $2$, $3$, $4$, and $c$ is a constant. Find the value of $P\left(\dfrac{1}{2} \lt \xi \lt \dfrac{5}{2}\right)$. Express your answer as a simplified fraction. | \dfrac{5}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
There are 18 teams participating in the opening ceremony of a competition. When entering, the 1st team has 27 members, the 2nd team has 26 members, and the 18th team has 10 members. If they enter in a single file, and all 18 teams' members are assigned numbers from 1 to 333 in the order they enter, then how many teams ... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $v = (a_{n+1} - a_n, a_{n+1} - a_n)$, and $v$ is the direction vector of the line $y = x$. If $a_1 = 5$, calculate the sum of the first 10 terms of the sequence $\{a_n\}$. Express your answer as a single integer. | 95 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a decreasing function defined on $(0,+\infty)$. If for all $x, y \in (0,+\infty)$, it satisfies $f(xy) = f(x) + f(y) - 1$, then find the solution set of the inequality $f(\log_{2}x-1) > 1$. Express your answer as an interval. | (2, 4) | big_math | [
"Mathematics -> Algebra -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, where a_1 + a_2 = 2 and a_4 + a_5 = 4, find the value of a_10 + a_11. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\log_2 0.3$, $b=3^{0.2}$, $c=0.3^2$, determine the correct order of these numbers from smallest to largest. Express your answer as a sequence of variable names, using less-than (<) symbols to denote the order (e.g., a < c < b). | a < c < b | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, let M be the midpoint of side AB, and let P be a point on AB between A and M. Draw MD parallel to PC, intersecting BC at D. Calculate the ratio of the area of triangle BPD to the area of triangle ABC. Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with a radius of 2cm, find the central angle in radians of a sector with an area of 4cm^2. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference d!= 0, and a_5^2 + a_7^2 + 16d = a_9^2 + a_11^2, calculate the sum of the first 15 terms S_15 of the sequence. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Four students, A, B, C, and D, signed up to participate in a $4 \times 100$ relay race. Let event $A$ be "Student A runs the first leg" and event $B$ be "Student B runs the second leg." Given that event $A$ has occurred, calculate the conditional probability $P(B|A)$. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the Xiaoshan taxi fare standard, where the starting price is 6 yuan (within 2 kilometers), 2.4 yuan per kilometer between 2 and 6 kilometers, and 3.6 yuan per kilometer for distances beyond 6 kilometers, calculate the total fare required for a passenger riding a taxi for a distance of x kilometers (x > 6). Expres... | 3.6x - 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
If $991+993+995+997+999=5000-N$, then find the value of $N$. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of ordered triplets of non-zero real numbers \((a, b, c)\) that satisfy the condition that each number is the product of the other two numbers. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In order to participate in outdoor expansion games, a class is divided into three teams: red, blue, and yellow. If students A and B independently choose a team at random, what is the probability that they choose the same team? Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations in two variables x and y:
\[
\left\{
\begin{array}{l}
x + 3y = 2k + 1 \\
x - y = 1
\end{array}
\right.
\]
and the solutions are opposite to each other, calculate the value of k. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the figure shown, $PQRS$ has three sides of equal length and $SR = 16 \text{ cm}$. If the perimeter of $PQRS$ is $40 \text{ cm}$, then what is the length of $PQ$? Express your answer in centimeters as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Marie changed her dice by replacing 1, 3, and 5 with -1, -3, and -5 respectively. She left the even numbers unchanged. If she throws two such dice, what is the total that cannot be achieved? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the distance between the line $x+y-1=0$ and the line $x+y+1=0$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
If a complex number z satisfies z + |z| = 2 + 8i, then compute |z|^2. Express your answer as a single integer. | 289 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_{n}} where all terms are positive numbers, and for any positive integers p and q, a_{p+q} = a_{p} * a_{q}. If a_{6} = 27, calculate the value of a_{10}. Express your answer as a single numerical value. | 243 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a batch of electronic products, the size of these products follows a normal distribution N(1, 4). If one product is randomly selected, calculate the probability that its size falls within the interval (3, 5). Express your answer as a decimal value between 0 and 1. | 0.1359 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the probability distribution of the random variable $\xi$ as $P(\xi=k)= \frac{1}{2^{k}}$, where $k=1$, $2$, $...$, determine the value of $P(2 < \xi\leqslant 4)$. Express your answer as a simplified fraction. | \frac{3}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.84375 | 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"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{3^{2011}+3^{2011}}{3^{2010}+3^{2012}}$. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function f(x) = ax^2 + bx + c, where a is a positive integer, c is greater than or equal to 1, and a + b + c is greater than or equal to 1, and the equation ax^2 + bx + c = 0 has two distinct positive roots both less than 1, find the minimum value of a. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of a geometric sequence {a_n} is S_n, if S_4 = 1, S_12 = 13, calculate the value of a_13 + a_14 + a_15 + a_16. | = 27 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$
\sqrt{2-\sqrt{3}} \cdot \sqrt{2-\sqrt{2-\sqrt{3}}} \cdot \sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}
$$
Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In polar coordinates, the curve $\rho = 4\cos \theta$ represents a circle. Find the maximum distance between any two points on this circle. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular vegetable garden is enclosed by a fence of length L, with one side against a wall. Calculate the maximum area of the garden in terms of L. | \dfrac{L^2}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, a_7 = 12. Find the value of a_3 + a_11. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A printer used 3289 digits to label the pages of a book. Calculate the total number of pages in the book. Express your answer as a whole number. | 1099 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If $\dfrac{x^{2}}{1-2k}-\dfrac{y^{2}}{k-2}=1$ represents a hyperbola with foci on the $y$-axis, find the range of the real number $k$. Express your answer as an interval. | (\frac{1}{2},2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex numbers $z_1$ and $z_2$ are symmetric about the imaginary axis in the complex plane, and $z_1 = 3 - i$, find the value of $\frac{z_1}{z_2}$. Express your answer as a complex number in the form $a + bi$. | -\frac{4}{5} + \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the linear function y = x - 1 passes through the point (m, 2), find the value of m. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
From five pairs of boots with distinct numbers, if we randomly select four boots, find the number of ways in which exactly one pair is included. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function f(x) = {2x-1 for x >= 2, -x^2 + 3x for x < 2}, calculate the value of f(-4) + f(4). Express your answer as a single integer. | -21 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
An electronic cat runs a lap on a circular track with a perimeter of 240 meters. It runs at a speed of 5 meters per second for the first half of the time and 3 meters per second for the second half of the time. Calculate the time it takes for the electronic cat to run the last 120 meters. Express your answer in seconds... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The product of four different positive integers is 360. What is the maximum possible sum of these four integers? Express your answer as a single integer. | 66 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\frac{x^2}{2}-\frac{y^2}{6}=1$, calculate the eccentricity of the hyperbola. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with the sum of the first n terms denoted as S_n, a_1 + a_3 = 5/2, and a_2 + a_4 = 5/4, calculate the expression S_n / a_n in terms of n. | 2^n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, a point (m, n) undergoes two translations: first, it is translated 2 units to the right, and then it is translated 1 unit upwards. Find the resulting coordinates of the point after these translations. Express your answer as an ordered pair (x, y). | (m+2, n+1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. If $3a=2b$, calculate the value of $\frac {2\sin ^{2}B-\sin ^{2}A}{\sin ^{2}A}$. Express your answer as a single number. | \frac{7}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{2}(5-x), & x\leq 1 \\ f(x-1)+1, & x>1 \end{cases}$, calculate the value of $f(2016)$. Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The line with equation \( y = x \) is translated 3 units to the right and 2 units down. Find the \( y \)-intercept of the resulting line. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Using only $2 coins and $5 bills, in how many different ways can you make $207? Express your answer as a single integer, representing the count of unique combinations of coins and bills that sum to $207. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1 + a_7 = 22 and a_4 + a_10 = 40. Find the common difference d. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Express 1.097 billion in scientific notation. Use the format a × 10^n, where 1 ≤ a < 10 and n is an integer. | 1.097 × 10^9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | 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. Find 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 | SynthLabsAI/Big-Math-RL-Verified |
For the fraction $\frac{1}{{x-5}}$ to be meaningful, what is the range of values for $x$? Express your answer as an inequality or a statement about the values of $x$. | x \neq 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the number 33 million into scientific notation. Express your answer in the form $a \times 10^{b}$. | 3.3 \times 10^{7} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given \(a > 0\), \(b > 0\), two lines \(l_1:(a-1)x+y-1=0\) and \(l_2:x+2by+1=0\), and \(l_1\) is perpendicular to \(l_2\), find the minimum value of \(\dfrac{2}{a}+\dfrac{1}{b}\). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the directrix equation of the parabola given by $x^{2}=4y$. Express your answer as an equation of a line in the form $y = c$. | y = -1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $\left(m-1\right)x^{2}+3x-1=0$, find the condition on $m$ such that the equation remains quadratic. Express your answer as an inequality or a condition on $m$. | m \neq 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
For every natural number n, the parabola y = (n^2 + n)x^2 - (2n + 1)x + 1 intersects the x-axis at points A_n and B_n. Let |A_n B_n| denote the distance between these two points. Calculate the value of |A_1 B_1| + |A_2 B_2| +... + |A_{1992} B_{1992}|. Express your answer as a fraction. | \frac{1992}{1993} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the sequence of positive terms $\{a_n\}$ satisfies $\lg a_{n+1} = 1 + \lg a_n$, and $a_{2001} + a_{2002} + a_{2003} + \dots + a_{2010} = 2013$. Calculate the value of $a_{2011} + a_{2012} + a_{2013} + \dots + a_{2020}$. Express your answer in the form $N \cdot 10^M$, where $N$ and $M$ are integers. | 2013 \cdot 10^{10} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_2 = 18 and a_4 = 8. Find the common ratio of the sequence. Express your answer as a fraction, which may be positive or negative. | \pm\frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $f(x) = f'(1)x^2 + x + 1$, calculate the definite integral $\int_{0}^{1} f(x) \, dx$. Express your answer as a fraction. | \frac{7}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin 68^\circ \sin 67^\circ - \sin 23^\circ \cos 68^\circ$ using trigonometric identities. Express your answer as a fraction or a decimal value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If acute angle $α$ satisfies $\sin \left(α+ \frac {π}{2}\right)= \frac {3}{5}$, calculate the value of $\sin α$. Express your answer as a simplified fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of $2^{-1}$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the initial function y = f(x) with graph c, reflect the graph about the line x = 1 to obtain graph c1, and then shift c1 one unit to the left to get graph c2. Express the function corresponding to graph c2 in terms of x. | y = f(1 - x) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overset{→}{e_1}, \overset{→}{e_2}$ are two unit vectors with an angle of $90^{\circ}$ between them, find the angle between $\overset{→}{a}=3\overset{→}{e_1}-\overset{→}{e_2}$ and $\overset{→}{b}=2\overset{→}{e_1}+\overset{→}{e_2}$. Express your answer in degrees. | 45 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{1,2,3,5,7\}$ and $N=\{x|x=2k-1, k \in M\}$, find the intersection $M \cap N$. Express your answer as a set, with elements separated by commas and enclosed in curly brackets. | \{1,3,5\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then find the length of edge $CD$. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In January 2000, the price of a kilogram of chicken was R$1.00, and it started to triple every 6 months. Calculate the time it will take for the price to reach R$81.00. Express your answer in years, rounded to 1 decimal place if necessary. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a}=(1,\cos \alpha)$ and $\vec{b}=(\sin \alpha,1)$, and knowing that $\vec{a} \perp \vec{b}$, calculate the value of $\sin 2\alpha$. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the roots of the quadratic equation x^{2}+px+q=0 are x_{1}=2 and x_{2}=-3, transform the original equation into the form (x - x_{1})(x - x_{2}) = 0 using the given roots. Express your answer in this exact format. | (x - 2)(x + 3) = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ by evaluating the polynomial at $x=1$ and $y=1$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function and $f(x-1)$ is an odd function, and $f(0.5) = 3$, find the value of $f(3.5)$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the height of a cylinder is 1, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of 2, calculate the volume of the cylinder. Express your answer in terms of π. | \frac{3\pi}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $ \cos(\pi + \alpha) = -\frac { \sqrt {10}}{5} $ and $\alpha \in \left(-\frac {\pi}{2}, 0\right)$, find the value of $\tan \alpha$. Express your answer as a simplified fraction, including any necessary square roots. | -\frac{\sqrt{6}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2xf'(1), calculate the value of f(1). Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the complex conjugate $\bar{z}$ of the complex number $z = \left(\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)^2$. Express your answer as a complex number in the form $a + bi$. | -\frac{1}{2} + \frac{\sqrt{3}}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^{\lg x} = a, where a is a constant and a > 1, find the product of all real roots of the equation. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left( n \right)={n}^{2}\cos \left( n\pi \right)$, and ${a}_{n}=f\left( n \right)+f\left( n+1 \right)$, calculate the value of ${a}_{1}+{a}_{2}+...+{a}_{100}$. Express your answer as a single integer. | -100 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the monomials $a^{m-1}b^{2}$ and $\frac{1}{2}ab^{n+5}$ is also a monomial, calculate the value of $n^{m}$. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of non-zero real number pairs (a, b) that satisfy the equation 1/a + 1/b = 1/(a+b). Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The second, third, and seventh terms of an arithmetic sequence $\{a_n\}$, which has a non-zero common difference, are exactly three consecutive terms of a geometric sequence $\{b_n\}$. Find the common ratio of $\{b_n\}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a_4$ and $a_{10}$ are the roots of the equation $x^2 - 8x + 1 = 0$ in an arithmetic sequence $\{a_n\}$, calculate the sum of the first 13 terms, $S_{13}$. Express your answer as a single numerical value. | 52 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(x_0) = 4$, calculate the limit: $\lim_{{\Delta x} \to 0} \frac{f(x_{0} + 2\Delta x) - f(x_{0})}{\Delta x}$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Three family members need to randomly select three different colored masks from five colors: blue, white, red, black, and green. Calculate the probability that both blue and white masks are selected. Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 1, 0)$ and $\overrightarrow{b} = (-1, 0, 2)$, and $k \overrightarrow{a} + \overrightarrow{b}$ is perpendicular to $2 \overrightarrow{a} - \overrightarrow{b}$, calculate the value of $k$. Express your answer as a single number. | \frac{7}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression $\dfrac{2i}{1-i}$ into the form $a+bi$, where $a$ and $b$ are real numbers. | -1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ∆ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. If {a}^{2}-{b}^{2}= \sqrt{3}bc and \sin C=2 \sqrt{3}\sin B, then find the measure of angle A in degrees. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $\angle A=60^{\circ}$, $\angle B=45^{\circ}$, and $BC=3\sqrt{2}$, calculate the length of side $AC$. Express your answer in simplest radical form. | 2\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Sylvia chose positive integers \( a, b \) and \( c \). Peter determined the value of \( a+\frac{b}{c} \) and got an answer of 101. Paul determined the value of \( \frac{a}{c}+b \) and got an answer of 68. Mary determined the value of \( \frac{a+b}{c} \). Calculate the value of \( \frac{a+b}{c} \) (which is equal to \( ... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_n$ is the sum of the first $n$ terms of the sequence ${a_n}$, and $a_1= \frac {1}{2}$, $a_{n+1}=1- \frac {1}{a_{n}}$, find the value of $S_{10}$. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Through the point (2,0), how many tangent lines can be drawn to the curve y = x^2 * e^x? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Equilateral triangle ABC has sides of length 4. The midpoint of BC is D, and the midpoint of AD is E. Calculate the value of EC^2. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A circle with its center at point M on the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$$ (where $a > 0$, $b > 0$) is tangent to the x-axis exactly at one of the hyperbola's foci F, and intersects the y-axis at points P and Q. If $\triangle MPQ$ is an equilateral triangle, calculate the eccentricity of t... | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms as S_n, if a_2 + a_4 + a_6 = 12, find the value of S_7. Express your answer as a single numerical value. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $i\\cdot z=z+a\\cdot i$ (where $i$ is the imaginary unit) and $|z|=\\sqrt {2}$, find the value of the positive number $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A 3-foot-by-5-foot poster needs to be completely covered with 3-inch-by-5-inch photos. How many photos will it take to cover the entire surface of the poster? Express your answer as a whole number. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function f(x) = sin(x) * (4cos^2(x) - 1). Express your answer in radians as a fraction of pi, if applicable. | \dfrac{2\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In 2004, Gerry downloaded 200 songs. In 2005, Gerry downloaded 360 songs at a cost per song which was 32 cents less than in 2004. Gerry's total cost each year was the same. Calculate the total cost of downloading the 360 songs in 2005. Express your answer in dollars, rounded to two decimal places if necessary. | 144.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of \( 129(\sqrt{3} x+\sqrt[3]{2})^{100} \), how many terms have rational coefficients? Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
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