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 the function f(n) = i^n - i^(-n) where n is a positive integer, find the number of elements in the set {f(n)}. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of positive integers {a_n} that satisfies the property a_{p+q} = a_p + a_q for any p, q in N*, and given that a_2 = 4, find the value of a_9. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(3,-2)$ and $\overrightarrow{b}=(x,-2)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(3,4)$ and $\overrightarrow{b}=(\sin \alpha,\cos \alpha )$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, calculate the value of $\tan \alpha$. Express your answer as a fraction in the form $\boxed{\frac{a}{b}}$. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the annual profit function y = -1/3x^3 + 81x - 234, where x is the annual output in ten thousand units, determine the annual output that will maximize the manufacturer's annual profit. Express your answer in ten thousand units. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation \(\sqrt{4 x^{2}+2 x+1}-\sqrt{4 x^{2}+14 x+5}=6 x+2\) and determine the number of real roots. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric sequence {a_n}, where a_1 = 1 and a_5 = 9, find the value of a_3. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1 = 2, and a_3 + a_5 = 10. Find the value of a_7. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The magnitudes $m_1$ and $m_2$ of two celestial bodies and their luminosities $E_1$ and $E_2$ are related by the equation $m_2 - m_1 = \frac{5}{2} \log_{10}{\frac{E_1}{E_2}}$. Given that the magnitude of the sun is $-26.7$ and the magnitude of Sirius (the brightest star in the night sky) is $-1.45$, find the ratio of t... | 10^{10.1} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given sets \( A = \{ x \mid 5x - a \leq 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{ 2, 3, 4 \} \), find the number of integer pairs \((a, b)\) that satisfy these conditions. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Express the quadratic equation $x^{2}-8x+3=0$ in the form $(x-4)^{2}=k$ and find the value of $k$. Provide your answer as a single number. | 13 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let X be a normally distributed random variable with mean μ = 2 and variance σ^2. Given that P(2 < X < 4) = 0.3, find the probability P(0 < X < 4). Express your answer as a decimal value between 0 and 1. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} is an arithmetic sequence with an even number of terms. The sum of its odd-numbered terms is 24, and the sum of its even-numbered terms is 30. If its last term is 21/2 greater than its first term, find the number of terms in the sequence. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality $ax^2+bx-2>0$ is $\{x|-2<x<-\frac{1}{4}\}$, calculate the value of $a+b$ using Vieta's formulas. Express your answer as a single integer. | -13 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select a number x in the interval [0,π]. Find the probability that -√3/2 < cos(x) < √3/2. Express your answer as a fraction or decimal. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Express $214.7$ billion in scientific notation. Provide your answer in the form $\boxed{a \times 10^b}$, where $a$ is a number between $1$ and $10$, and $b$ is an integer. | 2.147 \times 10^{11} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
If the product of $(x-m)$ and $(x-3)$ does not contain a linear term of $x$, then find the value of $m$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coordinates of the center of the circle C given by the equation $x^2 + y^2 - 2x + 2y - 2 = 0$. Express your answer as an ordered pair $(a, b)$. | (1, -1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1 + a_2 + a_3 + a_4 + a_5 = 20. Determine the value of a_3. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\left(1+ai\right)i=3+i$, where $i$ is the imaginary unit and $a$ is a real number, find the value of $a$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The polynomial 2x^4 - 3x^3 + ax^2 + 7x + b can be divided by x^2 + x - 2. Find the value of a/b. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(x = 4\) and \(x + y = 0\), find the value of \(y\). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Five people stand in a row, and there is exactly one person between person A and person B. Calculate the total number of possible arrangements. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If February is a month that contains Friday the 13th, what is the day of the week for February 1? Express your answer as the full day of the week. | Sunday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Five students (2 girls and 3 boys) are taking a photo. Calculate the number of different arrangements such that the two girls are not next to each other. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x + y + z = 1 \), \( x + y - z = 2 \), and \( x - y - z = 3 \), find the value of \( x \cdot y \cdot z \). Express your answer as a single numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let the sequence $\{a_n\}$ be a monotonically increasing arithmetic sequence, with $a_1=2$ and $a_1-1$, $a_3$, $a_5+5$ forming a geometric sequence. Calculate the value of $a_{2017}$. Express your answer as a single integer. | 1010 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Three people, A, B, and C, each hold some money. A says to B and C: "Each of you give me half of the money you hold, and then I will have 90"; B then says to A and C, "Each of you give me half of the money you hold, and then I will have 70"; C also says to A and B, "Each of you give me half of the money you hold, and t... | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of the real number $a$ if the coefficient of $x^3$ in the expansion of $\left(x- \frac {a}{x}\right)^{5}$ ($x \in \mathbb{R}$) is 10. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(2x) + sqrt(3)*cos(2x), find the sum of its zeros in the interval [0, pi]. Express your answer in terms of pi. | \frac {7\pi}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of (x+1+1/x)^6. Express your answer as a single integer. | 141 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets S={x|x²+2x=0, x∈R} and T={x|x²-2x=0, x∈R}, find the intersection of sets S and T, S∩T. Express your answer as a set, with elements separated by commas if necessary. | \{0\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m$ is a root of the equation $x^2 + x - 1 = 0$, find the value of the expression $3m^2 + 3m + 2006$. Express your answer as a single integer. | 2009 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The area of a trapezoidal field is 1400 square yards, and its altitude is 50 yards. Find the number of possible pairs of bases if each base is an integer divisible by 8. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle of inclination for line $l$ is 60°, calculate the slope of line $l$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, when $x > 0$, $f(x)= \begin{cases} ax+\log_{5}x, & x > 4 \\ x^{2}+2^{x}+3, & 0 < x\leqslant 4\\ \end{cases}$. If $f(-5) < f(2)$, calculate the range of values for $a$. Express your answer as an interval in standard interval notation. | (-\infty,2) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, calculate the value of $x$ in terms of $a$. | 27a^2 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $(1-i)z=|3-4i|$, determine the real part of $z$. Express your answer as a single fraction or decimal in the format $\boxed{[answer]}$. | \frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_1: ax+3y+1=0$ and $l_2: x+ay+2=0$ are perpendicular to each other, and $l_2$ is tangent to the circle $x^2+y^2=b$, calculate the value of $b$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle, two sides have lengths of $3$ and $8$ respectively. If the length of the third side is an even number, find the length of the third side that cannot be a valid option for the triangle. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.1875 | 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"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Methanol can be gently oxidized with hot copper metal. What is the product of this oxidation reaction? Express your answer as a chemical compound using standard chemical notation. | \text{HCHO} | big_math | [
"Mathematics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Each of the integers 1 to 7 is to be written, one in each circle in the diagram. The sum of the three integers in any straight line is to be the same. Find the number of different ways the centre circle can be filled. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A and B defined as:
\[ A = [-2, 4), \]
\[ B = \left\{ x \mid x^2 - ax - 4 \leq 0 \right\}. \]
Find the range of values for the real number a such that B ⊆ A. Express your answer as an interval. | [0, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{-1,0,1\}$, find the number of all non-empty proper subsets of the set $M$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A sequence $\{a_n\}$ is defined where $a_1=3$, $a_2=6$, and $a_{n+2}=a_{n+1}-a_n$. Calculate the fifth term of this sequence, $a_5$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given $(\bar{z} -1+3i)(2-i)=4+3i$ where $i$ is the imaginary unit, and $\bar{z}$ is the conjugate of $z$, calculate the imaginary part of $z$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows three rectangles and three straight lines. What is the value of \( p + q + r \) in degrees? | 180 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, when the line segment AB is translated to obtain the line segment A'B', if the coordinates of point A(-1,2) correspond to (3,-4) for point A', then what are the coordinates of point B' after applying the same translation to point B(2,4)? Express your answer as an ordered pair (x, y). | (6,-2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A geometric sequence \(\{a_{n}\}\) has the first term \(a_{1}=2^{-5}\). The geometric mean of its first 11 terms is \(2^{5}\). If removing one of the first 11 terms results in a new geometric mean of \(2^{4}\), which term number was removed? Provide your answer as the term number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Carl and André are running a race. Carl runs at a constant speed of x meters per second. André runs at a constant speed of y meters per second. André starts running 20 seconds after Carl begins. After André has been running for 10 seconds, he catches up to Carl. Calculate the ratio y : x. Express your answer as a ratio... | 3 : 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $ax-by-2=0$ and the curve $y=x^{3}$ have perpendicular tangents at point $P(1,1)$, calculate the value of $\frac {b}{a}$. Express your answer as a numerical value. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 3x^2 + 5x - 2, find the minimum value of the function f(x) in the interval [-2, -1]. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_2 = 3, and a_6 = 11, where S_n is the sum of the first n terms of the sequence {a_n}, find the value of S_7. Express your answer as a single numerical value. | 49 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ with $a > 0$ and $b > 0$, and its right focus $F$ at $(c,0)$ where $c = \\sqrt{a^2 + b^2}$, a perpendicular line is drawn from point $F$ to an asymptote of $C$, intersecting it at point $A$ and another asymptote at point $B$. If $2\\overrightarr... | \\dfrac{2\\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
When a fair die is rolled three times in succession, what is the probability of getting an odd number on the second roll? Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the inclination angle of the line $l: x+y+1=0$. Express your answer in degrees as a whole number. | 135 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the trapezoid shown in the given figure, \(AB\) is parallel to \(DC\), \(AD = AB = BC = 1 \text{ cm}\), and \(DC = 2 \text{ cm}\). Calculate the measure of the angle \(D \widehat{A} C\). Express your answer in degrees. | 90^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(m,-1)$ and $\overrightarrow{b}=(1,m+2)$, and knowing that $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $m$ that satisfies this condition. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ as the imaginary unit, calculate the product of the two complex numbers: $({\frac{1}{2}+\frac{{\sqrt{3}}}{2}i})({\frac{{\sqrt{3}}}{2}+\frac{1}{2}i})$. Express your answer in the form a+bi. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of four numbers is \( T \). Suppose that each of the four numbers is now increased by 1. These four new numbers are added together and then the sum is tripled. Express the value of this final result in terms of \( T \). | 3T + 12 | big_math | [
"Mathematics -> Algebra -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. Find the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$. Express your answer as a single real number. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A carpenter wishes to cut a wooden 3 x 3 x 3 cube into twenty-seven 1 x 1 x 1 cubes. What is the minimum number of cuts needed if he is allowed to rearrange the pieces after each cut? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: $a+b=1$, $a^{2}+b^{2}=3$, $a^{3}+b^{3}=4$, $a^{4}+b^{4}=7$, $a^{5}+b^{5}=11$. Identify the pattern and calculate $a^{7}+b^{7}$. Provide your answer as a single integer. | 29 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
How many ways can a student schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses can be taken in consecutive periods? Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the units digit of the value of $3^{1991} + 1991^{3}$ when expressed in decimal notation. Express your answer as a single digit. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left( \frac {\pi}{6}-\alpha \right)=\cos\left( \frac {\pi}{6}+\alpha \right)$, find the value of $\tan\alpha$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\dfrac{2}{1-\sqrt{3}i}$, calculate the modulus $|z|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
How many prime numbers p are there such that the expression 2p^4 - 7p^2 + 1 is equal to the square of an integer? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers z1 = 1 + i and z2 = 2 + bi (where b is a real number), find the value of b such that the product z1 * z2 is a real number. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an odd function with a domain of $\mathbb{R}$, and when $x \leq 0$, $f(x) = x^{2} - \frac{1}{2}x$. Find the numerical value of $f(1)$. Express your answer as a numerical value, including the negative sign if the answer is negative. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A cylindrical log has a length of 2 zhang and 4 chi, and a circumference of 5 chi. A kudzu vine starts growing at the base of the cylinder, wraps around the cylinder twice, and ends up even with the top of the log. What is the minimum length of the kudzu vine? Express your answer in chi as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x)=-x+log2((1-x)/(1+x))+1, calculate the value of f(1/2) + f(-1/2). | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let P be a point on the ellipse x^2/4 + y^2/3 = 1, and F1, F2 be the two foci of the ellipse. If angle F1PF2 = 60 degrees, calculate the dot product of PF1 and PF2. | 2 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = sin(wx) with w > 0, find the minimum value of w such that the function achieves its maximum value at least 50 times in the interval [0,1]. Express your answer as a numerical value, including π if necessary. | 100\pi | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
How many integers n satisfy the inequality n^4 - n^3 - 3n^2 - 3n - 17 < 0? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=\left( \frac{1}{2},k\right)$, $\overrightarrow{b}=\left(k-1,4\right)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of the real number $k$. Express your answer as a single real number. | \frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-2)^{2014} \div (-2)^{2013}$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Exponents"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence of letters KANGAROOKANGAROOKANG... the word KANGAROO is repeated indefinitely. What is the letter at the 2017th position in this sequence? Express your answer as a single letter. | K | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = x^2 and the line 2x - y + 1 = 0, find the coordinates of the point on the curve where the tangent line is parallel to the given line. Express your answer as an ordered pair (x, y). | (1,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial \(a_{0} x^{n}+a_{1} x^{n-1}+\cdots+a_{n-1} x+a_{n}\), where \(n\) is a non-negative integer, \(a_{0}\) is a positive integer, and the remaining \(a\) are integers or zero, let \(h = n + a_{0} + |a_{1}| + |a_{2}| + \cdots + |a_{n}|\). When \(h = 3\), how many polynomials satisfy this equation? Expre... | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations x + y = 2/3 and x/y = 2/3, calculate the value of x - y. Express your answer as a fraction. | -\frac{2}{15} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, $c > 0$ and the equation $a(a+b+c) + bc = 4$, find the minimum value of $2a+b+c$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = 2x^2 - 1 at point P(-3, 17). Express your answer in slope-intercept form (y = mx + b). | y = -12x - 19 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 125,000 in scientific notation. Provide your answer in the format $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 1.25 \times 10^{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Two perpendicular chords in a circle are divided into segments of lengths 3 and 4 on one chord and lengths 6 and 2 on the other chord. Find the diameter of the circle. Express your answer in exact form, possibly involving square roots. | \sqrt{65} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Box P has p chocolates and box Q has q chocolates, where p and q are both odd and p > q. Find the smallest number of chocolates which would have to be moved from box P to box Q so that box Q has more chocolates than box P. Express your answer as a formula in terms of p and q. | \frac{p - q + 2}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-2x+b=0$, find the sum of its roots $x_{1}+x_{2}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The weight of a sesame seed is approximately 0.00000201 kilograms. Convert this number to scientific notation. Express your answer in the format a × 10^b, where a is the coefficient and b is the exponent. | 2.01 × 10^(-6) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = (x - 1)^2 + 1, apply a 2-unit shift to the left and then a 3-unit shift down. Express the resulting function in the form y = f(x), including all necessary algebraic manipulations. | y = (x + 1)^2 - 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Three villages are connected by paths as follows:
1. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path.
2. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path.
3. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, if a_3 = 9 and a_5 = 1, find the value of a_4. Express your answer as a single value or with the ± symbol if necessary. | \pm 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = -n^2 + 8n$ ($n \in \mathbb{N}^*$), find the value of $n$ that maximizes $S_n$. Express your answer as a single integer value. | 4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of terms in the expansion of (x+y+z)^4. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{3} = 1$ ($a > 0$) has an eccentricity of $2$, calculate the real number $a$. Express your answer as a real number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, where a_3 + a_6 = 6 and a_5 + a_8 = 9, find the value of a_7 + a_10. Express your answer as a single value in the form \boxed{ }. | \frac{27}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-5x+2=0$, calculate the discriminant $\Delta$ using the formula $\Delta = b^{2}-4ac$, where $a=1$, $b=-5$, and $c=2$. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence with all terms positive, where $a_{1}a_{3}=16$ and $a_{3}+a_{4}=24$, calculate the value of $a_{5}$. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(\omega\) is a cube root of unity, calculate the value of \((1-\omega)\left(1-\omega^{2}\right)\left(1-\omega^{4}\right)\left(1-\omega^{8}\right)\). | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipses $C_{1}$: $\frac{x^2}{a^2}+y^{2}=1\left(a \gt 1\right)$ and $C_{2}$: $\frac{x^2}{4}+y^{2}=1$ with eccentricities $e_{1}$ and $e_{2}$, respectively, and the relationship $e_{2}=\sqrt{3}e_{1}$, calculate the value of $a$. Express your answer as a numerical value. | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.