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 equations x + lg x = 3 and x + 10^x = 3, find the sum of the roots x1 and x2 of these equations. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains two balls of the same size and shape, one red and one black. Drawing a ball randomly with replacement 3 times, each time drawing one ball. If drawing a red ball scores 2 points, and drawing a black ball scores 1 point, calculate the probability of scoring a total of 5 points from 3 draws. Express your an... | \frac {3}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the linear function $y=(m-1)x+(4m-3)$ lies in the first, second, and fourth quadrants. Find the range of $m$ that satisfies these conditions. Express your answer as an interval or inequality. | \left(\dfrac {3}{4}, 1\right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that R is the set of real numbers, A = {x | y = lg(x + 3)}, B = {x | x >= 2}, calculate the complement of A union B in R. Express your answer as a set definition using standard set notation. | { x | x <= -3 } | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x^2-1<0\}$ and $N=\{y|y=\log_{2}(x+2), x\in M\}$, calculate the intersection $M\cap N$. Express your answer as an interval or set, using any relevant mathematical notation. | (0,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 400 students at Pascal H.S., where the ratio of boys to girls is 3:2. There are 600 students at Fermat C.I., where the ratio of boys to girls is 2:3. What is the simplified ratio of boys to girls when considering all the students from both schools? Express your answer as a simplified ratio. | 12:13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A right triangle has one leg with a length of 11, and the lengths of the other two sides are also natural numbers. Calculate the perimeter of the triangle. Express your answer as a whole number. | 132 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$, where $a_1=2$ and $a_n=1-\frac{1}{a_{n-1}}$ for $n\geqslant 2$, find the value of $a_{2017}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (-2, 3)$, find the value of $k$ such that $k\overrightarrow{a} + \overrightarrow{b}$ is perpendicular to $\overrightarrow{a} - k\overrightarrow{b}$. Express your answer as a mathematical expression, including all possible solutions for $k$. | -1 \pm \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Meg starts with the number 100. She increases this number by 20% and then increases the resulting number by 50%. What is her final result? Express your answer as a numerical value. | 180 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest number of solid 2-in x 2-in x 1-in blocks that can fit in a 3-in x 2-in x 3-in box? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x \) and \( y \) are positive integers with \( x > y \) and \( x + xy = 391 \), calculate the value of \( x + y \). Express your answer as a single integer. | 39 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangular pyramid with three mutually perpendicular lateral edges of lengths 1, $\sqrt {3}$, and 2 respectively, and with all four vertices lying on the same spherical surface, calculate the volume of the sphere. Express your answer in terms of $\pi$. | \frac {8 \sqrt {2}}{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum difference between two positive integer factors of 1998. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\left\{0,1,2,3\right\}$ and $B=\left\{x\left|\right.x^{2}-2x-3\geqslant 0\right\}$, calculate the union of set $A$ and the complement of set $B$ in the real numbers, $A\cup \complement _{R}B$. Express your answer in interval or set notation. | (-1,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A certain university had 4,400 undergraduate students, 400 master's students, and 200 doctoral students applying to be volunteers. Using stratified sampling, 10 doctoral students were selected. Calculate the total number of volunteers selected from this university. Express your answer as a whole number. | 250 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \log_{2}x, & x > 1 \\ 3^{x}, & x\leqslant 1 \end{cases}$, calculate the value of $f(1)+f(2)$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the 998 natural numbers from 1000 to 1997 in any order in a row, and then sequentially calculate the sum of every three consecutive numbers. What is the maximum number of odd sums that can be obtained? Express your answer as a single integer. | 499 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence, and a_{n+1}+a_{n+2}=3n+5 (n∈ℕ*), find the value of a_1. Express your answer as a single value. | \dfrac{7}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse. Express your answer as a simplified fraction or decimal. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equation of a line:
$$
\begin{cases}
x = -1 + t\cos 50^\circ \\
y = -t\sin 50^\circ
\end{cases}
$$
where $t$ is the parameter, find the angle of inclination of the line. Express your answer in degrees as a whole number. | 130^\circ | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = 5x, find the expression of the new linear function obtained by translating it down 5 units. Express your answer as a linear equation in the form y = mx + b. | y = 5x - 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 346,000,000 in scientific notation. Use the format a × 10^b, where a is a number between 1 and 10, and b is an integer. | 3.46 × 10^8 | big_math | [
"Mathematics -> Other -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( Z_1, Z_2, \cdots, Z_{20} \) are the complex numbers corresponding to the 20 vertices of a regular 20-sided polygon inscribed in the unit circle in the complex plane, determine the number of distinct points corresponding to the complex numbers \( Z_1^{1995}, Z_2^{1995}, \cdots, Z_{20}^{1995} \). Express yo... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, $a, b \in \mathbb{R}$, and $(a+i)i=b-2i$, find the value of $a+b$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies $z=(3+4i)i$, calculate the real part of $z$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
If x is a positive number and lg x >= lg 2 + 1/2 lg x, find the minimum value of x. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given two functions f(x) and g(x) with domain and range both being the set {1,2,3}, defined as shown in the tables:
| x | 1 | 2 | 3 |
|---------|-----|-----|-----|
| f(x) | 2 | 1 | 3 |
| x | 1 | 2 | 3 |
|---------|-----|-----|-----|
| g(x) | 3 | 2 | 1 |
Find the solution set of the equation g[f(x)] = x + 1... | {1,2} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A shooter has a probability of 0.8 to hit the target with each shot. If he shoots 5 times in a row, calculate the probability that he hits the target exactly 4 times. Express your answer as a decimal value rounded to 4 decimal places. | 0.4096 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the trapezoid in the figure below, \(AB\) is parallel to \(CD\), \(AD = AB = BC = 1\) cm, and \(DC = 2\) cm. Find the measure of the angle \(\angle CAD\). Express your answer in degrees as a whole number. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If a_3 + a_{17} = 10, find the value of S_{19}. Express your answer as a single numerical value. | 95 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle between 0 degrees and 360 degrees that terminates in the same direction as -495 degrees. Express your answer in degrees, including the degree symbol. | 225^\circ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $z\left(3+i\right)=3+i^{2023}$, calculate the imaginary part of the conjugate complex number $\overline{z}$. Express your answer as a numerical value. | \frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Pat leaves Coe Hill at 1:00 p.m. and drives at a speed of 80 km/h for the first 60 km of the 150 km trip to Calabogie. How fast must he travel for the remainder of the trip to reach Calabogie by 3:00 p.m.? Express your answer in km/h. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $$\frac {7}{16} - \frac {7}{8}\sin^2 15°$$ using trigonometric identities. Express your answer as a simplified fraction, including any necessary square roots, in the form $$\boxed{[answer]}$$. | \frac {7 \sqrt {3}}{32} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
A school has selected 4 teachers from each of the first, second, and third grades of high school, as well as from the administrative department, to form a choir. Now, they need to select 3 leaders from these 16 people. The requirements are that these 3 people cannot all be from the same department, and at least 1 perso... | 336 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{2}x &,x>0 \\ \cos2\pi x &,x\leq 0 \end{cases}$, calculate the value of $f\left(\frac{1}{2}\right) + f\left(-\frac{1}{2}\right)$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$ with foci $F\_1$ and $F\_2$, and a point $P$ on the ellipse such that $ riangle PF\_1F\_2$ is a right triangle, find the area of $ riangle PF\_1F\_2$. Express your answer as a single number. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 23,766,400 in scientific notation. Provide your answer in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 2.37664 \times 10^{7} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a<sub>n</sub>} that satisfies a<sub>2</sub>+a<sub>4</sub>=4, a<sub>3</sub>+a<sub>5</sub>=8, find the sum of its first 8 terms. Express your answer as a single integer. | 40 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, where a_5 = 13 and S_5 = 35, find the common difference d. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\overrightarrow{a}| = 4$, $|\overrightarrow{b}| = 5$, and $|\overrightarrow{a} + \overrightarrow{b}| = \sqrt{21}$, calculate the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as an integer. | -10 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\sin\alpha,-3)$ and $\overrightarrow{b}=(1,\cos\alpha)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, calculate $\tan(\alpha+\frac{\pi}{4})$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $x-y+2=0$ and the circle $(C)$: $(x-3)^2+(y-3)^2=4$ (with center $C$), the line intersects the circle at points $A$ and $B$. Calculate the measure of $\angle ACB$ in degrees. | 90^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $Q$ is an intersection point of the ellipse $\frac{x^2}{49} + \frac{y^2}{b_1^2} = 1$ and the hyperbola $\frac{x^2}{16} - \frac{y^2}{b_2^2} = 1$, both sharing the same foci $F_1$ and $F_2$, find the value of $\frac{\lvert|QF_1| - |QF_2|
vert}{|QF_1| + |QF_2|}$. Express your answer as a simplified fracti... | \frac{4}{7} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(5,12), calculate the value of sin(α). Express your answer as a simplified fraction. | \frac{12}{13} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $P=\{x\in\mathbb{N}|1\leq x\leq 3\}$, and $Q=\{x\in\mathbb{R}|x^2+x-6=0\}$, find the intersection $P\cap Q$. Express your answer as a set, with elements separated by commas if necessary. | \{2\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of x^5 in the expansion of (x^2+x+1)^5. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integers k < 100 such that 2(3^(6n)) + k(2^(3n+1)) - 1 is divisible by 7 for any positive integer n. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
To understand the learning situation of 72 students, a systematic sampling method is used to extract a sample of size 8. Calculate the interval between segments. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the quadratic equation $x^{2}-8x+10=0$ into the form $(x+a)^{2}=b$ by completing the square. Express your answer in the form $(x+a)^{2}=b$. | $(x-4)^{2}=6$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function y = f(x) for any x1, x2 in R, and x1!= x2, the condition (x1-x2)[f(x1)-f(x2)] > 0 holds. If a = f(0.8^2), b = f(log2(0.8)), and c = f(2^0.8), what is the exact relationship between a, b, and c? Express your answer as a relationship between a, b, and c (e.g., a < b < c). | b < a < c | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $a+bi$ ($a, b \in \mathbb{R}$) corresponds to a point in the second quadrant of the complex plane, what are the conditions on $a$ and $b$? Express your answer using inequality notation. | a<0, b>0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(ABCD\) is a rectangle. If the area of triangle \(ABP\) is 40, calculate the area of the shaded region. Express your answer as a single number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? Express your answer in dollars and cents. | \$ 23.50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $$\frac {1-ai}{2+i}$$, where $a \in \mathbb{R}$ and $i$ is the imaginary unit, and the condition that the real part and the imaginary part of this complex number are equal, find the value of $a$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In a cube, count the number of lines containing the edges that are perpendicular to the line containing the edge AA₁. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a > 1 \) and \( b > 1 \) and \(\lg (a+b) = \lg a + \lg b\), calculate the value of \(\lg (a-1) + \lg (b-1)\). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: $(a+b=1)$, $(a^{2}+b^{2}=3)$, $(a^{3}+b^{3}=5)$, $(a^{4}+b^{4}=7…)$, and identify the pattern to find $(a^{10}+b^{10})$. Provide your answer as a single number. | 19 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Place five balls numbered 1 to 5 into five boxes also numbered 1 to 5, with each box holding one ball. If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $\sin 2\theta$ in the equation $\frac{\sqrt{2}\cos 2\theta}{\cos (\frac{\pi}{4}+\theta)}=\sqrt{3}\sin 2\theta$. Express your answer as a single fraction or decimal value. | -\frac{2}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_n = 4a_{n-1} + 3$, and $a_1 = 0$. Calculate the value of the fifth term of this sequence. Express your answer as a single integer. | 255 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the xy-plane, find the number of lines passing through the point (4,3) such that the x-intercept is a prime number and the y-intercept is a positive integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{$circle$\}$ and $B=\{$line$\}$, find the number of elements in $A\cap B$. Express your answer as a whole number. | 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sqrt[3]{9+ \frac{9}{m}}=9\cdot \sqrt[3]{ \frac{9}{m}}$, solve for $m$. Express your answer as a single integer. | 728 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest possible integer side length of a square that can be divided into 10 smaller squares, all with integer side lengths, and at least 8 of which have an area of 1. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, if $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=4$, and $\overrightarrow {a} + \overrightarrow {b}$ is perpendicular to $\overrightarrow {a}$, then find the angle between vector $\overrightarrow {a}$ and $\overrightarrow {b}$. Express your... | 120 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the integral equation $$\int_{0}^{1} (3x^2 + kx) \, dx = 3$$, calculate the value of k. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A sequence \( t_{1}, t_{2}, \ldots, t_{n}, \ldots \) is defined as follows:
\[
\begin{aligned}
& t_{1} = 14 \\
& t_{k} = 24 - 5 t_{k-1}, \text{ for each } k \geq 2
\end{aligned}
\]
For every positive integer \( n \), \( t_{n} \) can be expressed as \( t_{n} = p \cdot q^{n} + r \), where \( p, q \), and \( r \) are c... | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a convex quadrilateral ABCD, where AB is parallel to CD and AB + BC = CD + AD, determine the relationship between AD and BC. Express your answer using standard mathematical notation for relationships (e.g., =, <, >, cannot be determined). | AD = BC | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are both positive real numbers, find the maximum value of the expression x/(2x + y) + y/(x + 2y). Express your answer as a single fraction or decimal. | \frac{2}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the point corresponding to the complex number z is (-1,1). Calculate z(1+i) and express your answer in standard form. | -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\vec{a}$ and $\vec{b}$ are non-zero vectors and satisfy $(\vec{a}-2\vec{b})\perp\vec{a}$ and $(\vec{b}-2\vec{a})\perp\vec{b}$, find the angle between $\vec{a}$ and $\vec{b}$. Express your answer in radians as a fraction of π. | \dfrac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=f'(-2)e^{x}-x^{2}$, determine the value of $f'(-2)$. Express your answer as a fraction involving $e^2$. | \dfrac{4e^{2}}{e^{2}-1} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are rational numbers, and |x+5| + (y-5)^2 = 0, find the value of (x/y)^2023. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In the space Cartesian coordinate system, find the symmetric point of point P(1, 2, -3) with respect to the xOy coordinate plane. Express your answer as a coordinate (x, y, z). | (1, 2, 3) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial (x+a)^6, where a is a natural number (a ∈ N*), find the value of a for which the coefficient of the fourth term in the expansion is the largest. Express your answer as a single natural number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $Z$ satisfying $(2+i)Z=3-i$, find the magnitude of $Z$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the circle $x^2 + y^2 = 9$ is symmetric to the circle $x^2 + y^2 - 4x + 4y - 1 = 0$ with respect to line $l$, find the equation of line $l$ in the standard form $Ax + By + C = 0$. | x - y - 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle of inclination (slope angle) of the line $\sqrt {3}x+3y+1=0$. Express your answer in radians as a fraction of $\pi$. | \frac {5\pi}{6} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle ABC$ is a right-angled isosceles triangle. $D$ is the midpoint of $BC$ and $E$ is the midpoint of $AB$. If $AB = BC = 24 \text{ cm}$, find the area of $\triangle AED$. Express your answer in square centimeters (cm²) as a whole number. | 72 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin \alpha \cos \beta = -\frac{1}{2}$, find the range of values for $\cos \alpha \sin \beta$. Express your answer as an interval. | \left[-\frac{1}{2}, \frac{1}{2}\right] | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A bin contains 10 kg of peanuts. 2 kg of peanuts are removed and 2 kg of raisins are added and thoroughly mixed in. Then 2 kg of this mixture are removed and 2 kg of raisins are added and thoroughly mixed in again. Find the ratio of the mass of peanuts to the mass of raisins in the final mixture. Express your answer as... | 16:9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions f(x) = x + 4/x and g(x) = 2^x + a, if for all x1 in [1/2, 3], there exists x2 in [2, 3] such that f(x1) >= g(x2), then calculate the range of the real number a. Express your answer as an interval using standard interval notation. | (-\infty, 0] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function defined on R with f(2) = 0, and xf''(x) + f(x) < 0 holds true for all x > 0. Determine the solution set of the inequality xf(x) > 0 in interval notation. | (-2, 0) ∪ (0, 2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If \( S = 6 \times 10000 + 5 \times 1000 + 4 \times 10 + 3 \times 1 \), what is the exact value of \( S \)? Provide your answer as a single integer. | 65043 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations x/a + y/b + z/c = 1 and a/x + b/y + c/z = 0, find the value of the algebraic expression x^2/a^2 + y^2/b^2 + z^2/c^2. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $m=(t+1,1)$ and $n=(t+2,2)$, find the value of $t$ such that $m+n$ is perpendicular to $m-n$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A larger cube has a volume of 64 cm³. A smaller cube has edges that are half the length of the edges of the larger cube. Calculate the volume of the smaller cube. Express your answer in cubic centimeters (cm³) as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, if $a_1+a_2+a_3=32$ and $a_{11}+a_{12}+a_{13}=118$, find the value of $a_4+a_{10}$. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the real part of the complex number $Z= \frac{1-3i}{1+i}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac{x^2}{8} + \dfrac{y^2}{9} = 1$, find the distance between its two directrices. Express your answer as a single number. | 18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 4x, its axis intersects the x-axis at point A, and its focus is at point F. Line l passes through point A with a slope angle of π/3. Find the distance from point F to line l. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos (\pi-\alpha)= \frac {4}{5}$, and $\alpha$ is an angle in the third quadrant, calculate the value of $\tan 2\alpha$. Express your answer as a single fraction or decimal, including a negative sign if applicable. | \frac{24}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $\angle B=90^{\circ}$, $AC=4cm$, $BC=3cm$. Find the length of $AB$. Express your answer in centimeters, possibly involving a square root. | \sqrt{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation (x+3)(x-1) = 2x - 4, simplify and rewrite it in its general form ax^2 + bx + c = 0. | x^2 + 1 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The altitudes of a triangle are 12, 15, and 20. Find the largest angle in this triangle. Express your answer in degrees as a whole number. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A right triangle \( \triangle ABC \) has all side lengths as positive integers, and one of its legs is a root of the equation \( 2x^2 - 23x + 11 = 0 \). Calculate the area of this triangle. | 330 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of numbers 3, 5, 7, a, 4, where the average is 5, calculate the variance of this set of data. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The scores of 42 students in Class 8(1) are as follows.
| Score | 50 | 60 | 70 | 80 | 90 | 100 |
|-------|----|----|----|----|----|-----|
| Number of Students | 2 | 3 | 7 | 14 | 13 | 3 |
Find the median score of the students in this class. Express your answer as a single numerical value. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
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