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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|---|---|---|---|---|---|
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate... | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${{a_n}}$ that satisfies $a_1 = a_2 = a$, $a_n = a_{n-1} + a_{n-2}$, and $a_8 = 34$, find the value of $a$. Express your answer as a fraction in simplest form. | \dfrac{34}{21} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. Find the area of the border. Express your answer in square inches as a whole number. | 88 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a} = (-3, 2, 5)$ and $\overrightarrow{b} = (1, x, -1)$, and the dot product $\overrightarrow{a} \cdot \overrightarrow{b} = 2$, calculate the value of $x$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A point on the number line is 2021 units away from the origin. What is the numerical representation of this point? Express your answer using mathematical notation to indicate if there is more than one possible value. | \pm 2021 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Express 0.00124 grams per cubic centimeter in scientific notation. Provide your answer in the form $a \times 10^{-n}$. | 1.24 \times 10^{-3} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of the expression $1-(-2)-3-(-4)-5-(-6)$? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function y = 6/x, find the range of y values when x > 3. Express your answer in interval notation. | (0, 2) | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the integral $\int_{-1}^{2} |x| dx$. Express your answer as a single fraction or decimal value. | \frac{5}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_1 + a_2 = 3 and a_2 + a_3 = 6, find the value of a_7. Express your answer as a single integer. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ with positive terms, and $4a_1 - a_3 = 0$. Find the value of $\frac{S_3}{a_1}$. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a hyperbola as $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), a line with a slope of $\frac{\sqrt{3}}{3}$ passes through the left focus $F_1$ and intersects the right branch of the hyperbola at point $P$, and the y-axis bisects the line segment $F_1P$. Calculate the eccentricity of the ... | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
When tossing a uniform coin four times, what is the probability of getting at least two heads? Express your answer as a fraction in simplest form. | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that \( a \) and \( b \) are integers with \( 4 < a < b < 22 \). If the average (mean) of the numbers \( 4, a, b, 22 \) is 13, then find the number of possible pairs \( (a, b) \). Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let $C(A)$ denote the number of elements in a non-empty set $A$. Define
$$
A*B= \begin{cases}
C(A)-C(B), & \text{when } C(A) \geq C(B) \\
C(B)-C(A), & \text{when } C(A) < C(B)
\end{cases},
$$
If $A=\{x|x^2-ax-1=0, a\in \mathbb{R}\}$, $B=\{x||x^2+bx+1|=1, b\in \mathbb{R}\}$, let $S=\{b|A*B=1\}$. Calculate the number of... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Use the Euclidean algorithm to find the greatest common divisor of 456 and 357. Express your answer as a single whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval where the function $y=\frac{1}{2}x^2-\ln x$ is monotonically decreasing. Express your answer in standard interval notation. | (0,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Divide 12 people into 2 groups evenly, and then designate one person as the leader and another as the deputy leader from each group. What is the probability of A being chosen as the leader? Express your answer as a simplified fraction. | \frac {1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The five major classical Chinese operas are "The Peony Pavilion," "The Story of the Western Wing," "The Peach Blossom Fan," "The Injustice to Dou E," and "The Palace of Eternal Youth." If two of the five opera classics are selected, what is the probability of selecting exactly one of "The Peony Pavilion" and "The Story... | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A line l passing through the point (0,1) intersects the circle (x-1)^2 + y^2 = 4 such that the chord formed has the minimum length. Determine the slope of the line l. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Patrick is a math teacher for 15 students. After a quiz, he found that the average score of the other students, excluding Peyton's score, is 80 points. Including Peyton's score, the class's average score is 81 points. Calculate Peyton's score in this quiz. Express your answer as a whole number. | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ is the opposite number of $1$, find the absolute value of $x$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. Calculate the total amount of metal wasted as a fraction of the original square's area. | \frac{1}{2}\text{ the area of the original square} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the positive geometric sequence {a_n}, a_2 = 3 and a_8 = 27. Calculate the value of the fifth term a_5. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In a Cartesian coordinate system, a ray of light starts at point A (-3, 5), reflects off the x-axis, and then reaches point B (2, 7). Calculate the distance that the ray of light travels from A to B. Express your answer as a single number. | 13 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first n terms of an arithmetic sequence {a_n} be S_n. Given that S_3/S_6 = 1/3, calculate the value of S_6/S_12. Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x) = 2x + 1$ have a domain of $[1, 5]$. Find the domain of the function $f(2x - 3)$. Express your answer in interval notation, using square brackets for closed intervals. | [2, 4] | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows a circle with centre \( O \) and a triangle \( O P Q \). Side \( P Q \) is a tangent to the circle. The area of the circle is equal to the area of the triangle. Calculate the ratio of the length of \( P Q \) to the circumference of the circle. Express your answer as a simplified ratio in the form a:b. | 1:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 and a_{10} are the two roots of the equation x^2-x-6=0. Calculate the value of a_4 * a_7. Express your answer as an integer. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $i^{2013}$. Express your answer as a single value. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that there are 2 defective products in 6 items, calculate the probability of selecting exactly 1 defective product when taking 2 items at random. Express your answer as a simplified fraction. | \dfrac{8}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. Calculate the ratio of the perimeter of the rectangle to the perimeter of the rhombus. Express your answer as a ratio (e.g., 1:1, 2:1, etc.). | 1:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, let O be the origin (0, 0) and P be the point (6, 8). After rotating the vector $\overrightarrow{OP}$ counterclockwise by $\frac{\pi}{4}$, we obtain the vector $\overrightarrow{OQ}$. Find the coordinates of point Q. Express your answer as a coordinate pair (x, y). | (-\sqrt{2}, 7\sqrt{2}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression -10 + (-12). Provide your answer as a single integer. | -22 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation:
\[
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
\]
where \([x]\) denotes the greatest integer less than or equal to \(x\), find the smallest value of \(x\) that satisfies this equation. | 11.5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality -2x^2 + 5x - 2 > 0, evaluate the expression sqrt(4x^2 - 4x + 1) + 2|x - 2| and provide the exact value. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $\begin{cases} x=1-2t, \\ y=2+3t \end{cases}$ (where $t$ is a parameter) and the line $4x+ky=1$, calculate the constant $k$ if the two lines are perpendicular. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = e^(ax) with respect to x. Express your answer as a function of x. | ae^{ax} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If the complex number $z$ satisfies $(2-5i)z=29$, calculate the value of $z$. Express your answer in the form $a + bi$. | 2+5i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (6, x)$ and $\overrightarrow{b} = (2, -2)$, and that $(\overrightarrow{a} - \overrightarrow{b}) \perp \overrightarrow{b}$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest whole number that is larger than the sum of 2 1/2, 3 1/3, 4 1/4, and 5 1/5. Express your answer as a single whole number. | 16 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $a=7$, $b=8$, $\cos B=\frac{1}{7}$, calculate the size of $\angle A$ in radians. Express your answer using $\pi$. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The perimeter of a rectangle is 8, with one side being -a-2. Find the length of the other side. Express your answer as an algebraic expression involving the variable 'a'. | 6 + a | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + (a - 2)x^2 + 2x, where f(x) is an odd function, find the equation of the tangent line to the curve y = f(x) at the point (1, 3). Express your answer in slope-intercept form (y = mx + b). | y = 5x - 2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations",
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression (3+√8)^2n, where n is a natural number and n ≥ 2009, calculate the remainder when [(3+√8)^2n] is divided by 8. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A painting measuring 18 inches by 24 inches is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, calculate the ratio of the smaller to the larger dimension of the framed pai... | 2:3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x\left|\right.\left(x-2\right)\left(x+a\right)=0,a\in R\}$, $B=\{x∈N|\frac{6}{x-1}≥2\}$. Find the number of elements in $A\cup B$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
What is the minimum number of operations required to transform 2014 into 1, where an operation is defined as adding, subtracting, multiplying, or dividing by a single-digit number (1-9)? | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, x)$, $\overrightarrow{b} = (2-3x, 2)$, $\overrightarrow{c} = (-1, 2)$, if $\overrightarrow{a}$ is perpendicular to ($\overrightarrow{b} + \overrightarrow{c}$), find the value of $x$ that satisfies this condition. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all integers between 50 and 350 that end in 1. | 5880 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A semicircle with radius 1 is centered at 0 and has AB as its diameter. A point C moves along the entire circumference of the circle. Extend AC to point D so that CD/CB = k (where k > 0 and k is a constant). Calculate the length of the curve traced by point D. Express your answer in terms of pi and the square root of 1... | \pi \sqrt{1 + k^2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $8\\cdot2^x = 5^{y + 8}$, find the value of x when y = -8. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
All the students in an algebra class took a 100-point test. Five students scored 100 points each, each student scored at least 60 points, and the mean score was 76 points. Calculate the smallest possible number of students in the class. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{2+4i}{(1+i)^{2}}$. Express your answer as a simplified complex number. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 4 black balls and 3 white balls. Two balls are drawn without replacement. Given that the first ball drawn is black, calculate the probability of drawing a white ball on the second draw. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$. If $(n+1)S_{n} < nS_{n+1}$ for all $n \in \mathbb{N}^{*}$ and $\frac{a_{8}}{a_{7}} < -1$, find the maximum value of $n$ for which $S_{n} < 0$. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression $\dfrac{1}{(1+i)^{2}}$ (where $i=\sqrt{-1}$). Express your answer in standard form as a complex number. | -\dfrac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^{2}}{6} - \frac{y^{2}}{3} = 1$ with foci $F_1$ and $F_2$, and a point $M$ on the hyperbola such that $MF_1$ is perpendicular to the x-axis, find the distance from $F_1$ to the line $F_2M$. Express your answer as a single value. | \frac{6}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \(\frac{1+3+5}{10+6+2}\) and simplify the resulting fraction. Express your answer in the form \(\boxed{\frac{a}{b}}\), where \(a\) and \(b\) are integers and the fraction is in its simplest form. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A point $M$ is randomly chosen within a parallelogram $ABCD$ with an area of $S$. Calculate the probability that the area of $\triangle MCD$ is less than $\frac{S}{3}$. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, calculate the probability that the under-side is also red. Expres... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Let $e_1$ and $e_2$ be the eccentricities of an ellipse and a hyperbola, respectively, both sharing common foci $F_1$ and $F_2$. Let $P$ be a common point on both curves such that $| \overrightarrow{PF_1} + \overrightarrow{PF_2}|=| \overrightarrow{F_1F_2}|$. Calculate the value of $\frac{e_1e_2}{\sqrt{e_1^2 + e_2^2}}$.... | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x) = x^{2} - 4x + 4$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A triangular prism has a volume of 120 cm³. Two edges of the triangular faces measure 3 cm and 4 cm, as shown. Calculate the height of the prism in centimeters. | 20 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation A + B = 5, find the value of the expression B - 3 + A. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $a+b=1$, $a^{2}+b^{2}=3$, $a^{3}+b^{3}=4$, $a^{4}+b^{4}=7$, $a^{5}+b^{5}=11$, and observing that each term $a^{n}+b^{n}$ for $n \geq 3$ is the sum of the two preceding terms, calculate the value of $a^{7}+b^{7}$. Provide your answer as a single integer. | 29 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation e^a + 2a = e^b + 3b, where e is the base of the natural logarithm, a > 0, and b > 0, determine the relationship between a and b. Express your answer in the format 'a [relation] b' where [relation] is one of >, <, or =. | a > b | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
When the point (x,y) moves on the line x+y-3=0, find the minimum value of the expression 2^x + 2^y. Express your answer as a numerical value. | 4 \sqrt{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ corresponds to the point $(2,1)$ in the complex plane, calculate the value of $\frac{{2i}}{{z-1}}$. Express your answer as a complex number in the form $a+bi$. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(x^{2} + x - 1 = 0\), calculate the value of the expression \(2x^{3} + 3x^{2} - x\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola x^2/4 - y^2/m = 1 with an eccentricity equal to the radius of a circle that is centered at the right focus and is tangent to the asymptote of the hyperbola, determine the value of m. Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A certain school held a campus singer competition. The scores of $5$ contestants are $9$, $8.7$, $9.3$, $x$, and $y$. It is known that the average score of these $5$ contestants is $9$, and the variance is $0.1$. Calculate the absolute difference between $x$ and $y$, i.e., $|x-y|$. | 0.8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the geometric mean of 4 and 9. Express your answer as ±value. | \pm 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(PW\) is parallel to \(QX\). \(S\) and \(T\) lie on \(QX\), and \(U\) and \(V\) are the points of intersection of \(PW\) with \(SR\) and \(TR\), respectively. If \(\angle SUV = 120^\circ\) and \(\angle VTX = 112^\circ\), calculate the measure of \(\angle URV\). Express your answer in degrees (°). | 52^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The interior angles of a triangle are \((5x + 3y)^\circ\), \((3x + 20)^\circ\), and \((10y + 30)^\circ\), where \(x\) and \(y\) are positive integers. Using the fact that the sum of the interior angles of a triangle is \(180^\circ\), calculate the value of \(x + y\). Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
What is the decimal number equivalent to the binary number $110_{(2)}$? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. Find the next palindrome greater than 13931 and calculate the sum of its digits. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. Express your answer as a formula or expression in terms of $n$. | n(n+2) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( \sqrt{a(x-a)} + \sqrt{a(y-a)} = \sqrt{x-a} - \sqrt{a-y} \) holds true within the range of real numbers, where \( a, x, y \) are three distinct real numbers, find the value of \( \frac{3x^2 + xy - y^2}{x^2 - xy + y^2} \). Express your answer as a single fraction or integer. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $a + \\frac{1}{a} = 7$, calculate the value of $a^{\\frac{1}{2}} + a^{-\\frac{1}{2}}$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Divide the numbers 1, 2, 3, …, 19 into several groups, with at least one number in each group, so that in each group containing more than two numbers, the least common multiple (LCM) of any two numbers is not in the same group. What is the minimum number of groups required? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she begins working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in 12-hour clock format (e.g., 4:10 P.M.). | 4:10 P.M. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Five positive integers are listed in increasing order. The difference between any two consecutive numbers in the list is three. The fifth number is a multiple of the first number. How many different such lists of five integers are there? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given $3^{m}=4$ and $3^{n}=6$, calculate the value of $3^{2m-n}$. Express your answer as a fraction. | \frac{8}{3} | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the given geometric setup, let \( I \) be the incenter and \( O \) be the circumcenter of the acute \(\triangle ABC\). Line \( OI \) intersects \( BC \) at point \( X \). Let \( M \) be the midpoint of the minor arc \(\overparen{BC}\) that does not include point \( A \) on the circumcircle of \(\triangle ABC\). Give... | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\frac{-1-2i}{(1+i)^{2}}$, simplify the expression for $z$ and express your answer in the form $a+bi$. | -1 + \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is an arithmetic sequence, and a_3 + a_{11} = 50, also a_4 = 13, find the value of a_2. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given segments of lengths $2$, $3$, $5$, $7$, and $11$, calculate the probability that three segments selected from these five cannot form a triangle. Express your answer as a simplified fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first n terms of an arithmetic sequence is 48, and the sum of the first 2n terms is 60. Calculate the sum of the first 3n terms. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Select 3 teachers from 4 male teachers and 3 female teachers to be sent to 3 different suburban schools for teaching, one teacher per school. The selection must include both male and female teachers. Calculate the total number of different selection and assignment plans. Express your answer as a single integer. | 180 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 4x - 5 with domain x ∈ [-3, 2], find the range of the function. Express your answer in standard interval notation (e.g., [a, b]). | [-9, 7] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $1 + 0.1 - 0.1 + 1$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(\left\{a_{n}\right\}\) with the general term formula \(a_{n}=\frac{2n-17}{2n-19}\) for \(n=1,2,\cdots\), find the largest term in the sequence. Express your answer as a specific term in the sequence (e.g., a_10). | a_{10} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation (x+3)(x-1) = 2x - 4, simplify and express it in its general form as a quadratic equation in the form ax^2 + bx + c = 0. | x^2 + 1 = 0 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \((\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}\). Express your answer in simplest radical form if necessary. | 3+2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with one of its asymptotes defined by the equation $y = \frac{\sqrt{5}}{2} x$, and it shares a common focus with the ellipse $\frac{x^{2}}{12} + \frac{y^{2}}{3} = 1$, derive the equation of the hyperbola $C$ in standard form. | \frac{x^{2}}{4} - \frac{y^{2}}{5} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The general term of the sequence {a_n} is a_n = n^2 + n. Find the sum of the first 10 terms of the sequence {1/a_n}. Express your answer as a simplified fraction. | 10/11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\frac{\pi}{2} < \alpha < \pi$ and $\sin \alpha + \cos \alpha = \frac{1}{5}$, calculate the value of $\frac{2}{\cos \alpha - \sin \alpha}$. Express your answer as a simplified fraction, including the negative sign if applicable. | -\frac{10}{7} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A student simultaneously throws two dice, and the numbers obtained are denoted as a and b, respectively. For the hyperbola x^2/a^2 - y^2/b^2 = 1, calculate the probability that the eccentricity e = sqrt(a^2 + b^2)/a > sqrt(5). Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-2x+3=0$, what is the coefficient of the linear term? Provide your answer as a numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
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