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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|---|---|---|---|---|---|
A person walks a distance of 378 li, walking briskly on the first day, and from the second day onwards, the distance walked each day is half of the previous day's distance. After walking for 6 days, the destination is reached. Find the distance walked on the second day. Express your answer in li as a whole number. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The organizing committee plans to allocate six staff members, named A, B, C, D, E, and F, to two different locations for reception work during the meeting. If A and B must be in the same group and each group must have at least two people, calculate the number of different distribution methods. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A school plans to divide 2 female teachers and 4 male teachers into 2 groups to participate in urban-rural exchange activities in locations A and B. Each group must consist of 1 female teacher and 2 male teachers. Calculate the total number of different arrangements that meet these specifications. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation of a circle $\rho=\sqrt{2}(\cos \theta+\sin \theta)$, find the coordinates of the center of this circle. Express your answer in polar coordinates as (r, θ). | \left(1, \frac{\pi}{4}\right) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if the complex number $i \cdot z = \sqrt{2} - i$, then find the imaginary part of the complex number $z$. Express your answer as a single number, including the 'i' symbol if it's an imaginary number. | -\sqrt{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
ABCD is a rectangle. P is any point on AB. PS is perpendicular to BD, PR is perpendicular to AC, AF is perpendicular to BD, and PQ is perpendicular to AF. Find the sum of PR and PS. Express your answer as a length (e.g., AF). | AF | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The polynomial equation 64x^3 - 144x^2 + 92x - 15 = 0 has roots that form an arithmetic sequence. Find the difference between the largest and smallest root. | 1 | big_math | [
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. Calculate the imaginary part of the complex number $z= \frac {i-1}{i+1}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given two arithmetic sequences {a_n} and {b_n}, whose sums of the first n terms are A_n and B_n respectively, and it is known that A_n/B_n = (4n+2)/(5n-5), calculate the value of (a_5+a_{13})/(b_5+b_{13}). Express your answer as a fraction in simplest form. | 7/8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $z=\frac{3+i}{2-i}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 80,000,000 in scientific notation as $8 \times 10^n$. What is the value of $n$? Provide your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Randomly split 2.5 into the sum of two non-negative numbers. Then, round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3? Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the cosine value of the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{3}{5}$, and $| \overrightarrow{a}|=| \overrightarrow{b}|=1$, find the magnitude of the sum of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a decimal value, including the squa... | \frac{4 \sqrt{5}}{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the circle $$(x-3)^{2}+(y+\frac{7}{3})^{2}=1,$$ find the coordinates of its center. Express your answer as an ordered pair (a, b). | (3, -\frac{7}{3}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\Delta ABC$, the coordinates of $A$ and $B$ are $(0, 2)$ and $(0, -2)$, respectively. If the perimeter of the triangle is $10$, then find the equation of the trajectory of vertex $C$. Express your answer in the standard form of an ellipse equation. | \dfrac{x^2}{5}+\dfrac{y^2}{9}=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Team A alone needs $m$ days to complete a project, and team B alone needs $n$ days. If team A and B work together to complete this project, calculate the required number of days in terms of $m$ and $n$. Express your answer as a formula. | \frac{mn}{m+n} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A student attempted a problem by calculating the sum of two polynomials A and B, but mistakenly treated it as the difference A - B. As a result, they obtained 9x^2 - 2x + 7. If B = x^2 + 3x - 2, then calculate the correct sum A + B. Express your answer as a polynomial in standard form, including all terms and coefficie... | 11x^2 + 4x + 3 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to the below conversation, $B:$ I'm a truth teller. $C:$ $D$ is a truth teller. $D:$ $B$ and $E$ are not both truth tellers. $E:$ $A$ and $B$ are truth ... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Five students stand in a circle and count in sequence. The rules are as follows: the first student starts counting from 2, the second student also starts counting from 2, and thereafter, each student's number is the sum of the previous two students' numbers. If the number called is a multiple of 3, the student must cla... | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The equations $2\sqrt{\frac{2}{3}}=\sqrt{2\frac{2}{3}}, 3\sqrt{\frac{3}{8}}=\sqrt{3\frac{3}{8}}, 4\sqrt{\frac{4}{15}}=\sqrt{4\frac{4}{15}}, 5\sqrt{\frac{5}{24}}=\sqrt{5\frac{5}{24}}$ follow a specific pattern. If the equation $8\sqrt{\frac{8}{n}}=\sqrt{8\frac{8}{n}}$ follows the same pattern, calculate the value of n. ... | 63 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with magnitudes $|\overrightarrow {a}| = 6\sqrt {3}$ and $|\overrightarrow {b}| = \frac {1}{3}$, respectively, and their dot product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, calculate the angle $\theta$ between $\overrightarrow {a}$ and $\overri... | \frac {5\pi}{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( p \) and \( q \) be two consecutive terms of the sequence of odd primes. What is the minimum number of positive divisors of \( p + q \)? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of ways to select 4 shoes from 5 pairs of different-sized shoes such that at least 2 of the 4 shoes can be paired together. Express your answer as a whole number. | 130 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that line segments a, b, c, and d are in proportion and a = 1, b = 2, c = 3, find the value of d. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan \left( \frac{\pi}{4}+α\right)=2$, calculate the value of $\frac{\sin α-\cos α}{\sin α+\cos α}$. Express your answer as a single fraction or decimal value. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 students signing up for two extracurricular activity groups, with each student limited to joining one of the groups. Calculate the total number of different sign-up methods. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {x | x - 1 ≤ 0} and set B = {0, 1, 2}, find A ∩ B. Express your answer as a set. | \{0, 1\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Define an operation $m \Delta n$: when both $m$ and $n$ are either positive even numbers or positive odd numbers, $m \Delta n = m + n$; when one of $m$ and $n$ is a positive odd number and the other is a positive even number, $m \Delta n = mn$. Let $M = \{(x,y) | x \Delta y = 36, x \in \mathbb{N}^*, y \in \mathbb{N}^*\... | 40 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
When $\alpha$ is an angle in the second quadrant, find the value of $\frac{|\sin \alpha|}{\sin \alpha} - \frac{|\cos \alpha|}{\cos \alpha}$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let $P= \log_{2}3$, $Q= \log_{3}2$, and $R= \log_{2}(\log_{3}2)$. Determine the correct inequality for the values of P, Q, and R. Express your answer as an inequality (e.g., R < Q < P). | R < Q < P | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real roots of the equation $(x+1)|x+1|-x|x|+1=0$. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 and a_5 are the roots of the equation x^2-10x+16=0. Given that a_1 and a_5 are positive, find the value of a_3. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A group of students went boating, but there weren't enough boats in the park. If each boat holds 4 people, there will be 10 people left without a boat. If each boat holds 5 people, there will be 1 person left without a boat. Calculate the total number of people in the group. Express your answer as a whole number. | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A group of 56 soccer fans from Chongqing took taxis from their hotel to the stadium. There are two taxi companies, A and B, with company A having 3 fewer taxis than company B. If all were arranged to take taxis from company A, with 5 people per taxi, there would not be enough taxis; with 6 people per taxi, some taxis w... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Hagrid has 100 animals. Among these animals:
- Each is either striped or spotted but not both.
- Each has either wings or horns but not both.
- There are 28 striped animals with wings.
- There are 62 spotted animals.
- There are 36 animals with horns.
Calculate the number of Hagrid's spotted animals that have horns. ... | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence ${\{(a_{n})\}}$ where each term is greater than zero, the first term $a_{1}$ equals 1, and the sum of the first $n$ terms $S_{n}$ satisfies $S_{n} \sqrt{{S}_{n-1}} -S_{n-1} \sqrt{{S}_{n}}=2 \sqrt{{S}_{n}{S}_{n-1}} (n \in N^{*}, n \geqslant 2)$, calculate the value of $a_{81}$. Express your answer as a ... | 640 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, if $a_3 + a_4 + a_5 + a_6 + a_7 = 450$, calculate the value of $a_2 + a_8$. Express your answer as a single integer. | 180 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Five friends went to the cinema to watch a movie, and their seats are arranged in a single row from left to right, numbered from 1 to 5. During the movie, Ada went to the hall to buy some popcorn. When she returned, she found that:
- Bea had moved two seats to the right.
- Cee had moved one seat to the left.
- Dee and... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a set $M \subseteq \{4, 7, 8\}$, where at most one even number can be present in set $M$, count the number of such sets $M$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of the real number 9. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x) = e^x - e^{-x}$, and $g(x) = \log(mx^2 - x + \frac{1}{4})$. If for any $x_1 \in (-\infty, 0]$, there exists $x_2 \in \mathbb{R}$ such that $f(x_1) = g(x_2)$, then find the minimum value of the real number $m$. Express your answer as a single real number. | -\frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
There are two wooden sticks with lengths of 3cm and 5cm respectively. Determine the range of lengths that can form a triangle with the two sticks. Express your answer in centimeters (cm). | (2, 8) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is a function defined on R, and f'(x) is the derivative of f(x), if f'(x) + f(x) > 0, and f(1) = 1/e, find the solution set of the inequality f(ln x) < 1/x. Express your answer as an interval. | (0,e) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(1+x^{2})(1- \frac {1}{x})^{6}$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^5$ in the expansion of $(1-x^3)(1+x)^{10}$. Express your answer as a single integer. | 207 | big_math | [
"Mathematics -> Algebra -> Polynomials -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, for a complex number $z$ satisfying $|z+2i|=|z|$, find the imaginary part of $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the average highest monthly temperatures (in degrees Celsius) in our city in 2022 were $12$, $11$, $10$, $20$, $23$, $28$, $36$, $36$, $31$, $24$, $23$, $19$, find the 60th percentile of this data set. Express your answer as a single number in degrees Celsius. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Dave walks to school at a rate of 90 steps per minute, with each step being 75 cm long. It takes him 16 minutes to get to school. His brother, Jack, going to the same school by the same route, averages 100 steps per minute, but his steps are only 60 cm long. Calculate the time it takes Jack to get to school. Express yo... | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions (P, Q) of the equation P/Q - Q/P = (P+Q)/PQ, where P and Q are integers from 1 to 9 inclusive. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2\cos ^{2}15^{\circ}-1$. Express your answer as a numerical value, including any necessary mathematical notation. | \dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, given $BC=6$, $AC=4$, $\sin A=\frac{3}{4}$, calculate the measure of angle $B$ in radians. Express your answer using $\pi$. | \frac{\pi}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^4$ term in the expansion of the binomial $(x^2 - \frac{1}{x})^5$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function f(x) = x^2 + mx + m, find the range of values for m such that the function is increasing on the interval (-2, +∞). Express your answer as an interval or inequality, using any necessary mathematical notation (e.g., ≥, ≤, ∞). | [4, +\infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( x \), \( y \), and \( z \) be non-negative numbers. Suppose \( x + y = 10 \) and \( y + z = 8 \). Let \( S = x + z \). Calculate the sum of the maximum and the minimum value of \( S \). Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ ($a \neq 0$) is a root of the equation $x^2 + bx - 2a = 0$, find the value of $a + b$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $a=(\log_{2}3)^3$, $b=\ln 2$, and $c=\dfrac{1}{\sqrt{5}}$, evaluate and order the expressions from least to greatest. Express your answer as a series of inequalities using the exact variable names (e.g., a < b < c). | c < b < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given a mapping from set A to B, $f: x \rightarrow y = 2x + 1$, calculate the image of element 2 in set A in set B. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $a$ and $b$ are positive real numbers, with $P=a^{a}b^{b}$ and $Q=a^{b}b^{a}$. Determine the relationship between $P$ and $Q$, expressing your answer using standard inequality notation. | P \geq Q | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A certain store sells washing machines of a certain brand. Each washing machine costs $a$ yuan to purchase. The store increases the cost by $30\%$ for selling, and then offers a $20\%$ discount. Calculate the retail price of the washing machine in terms of $a$. | 1.04a | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Beverly, Dianne, Ethan, and Jamaal want to sit in a row of four seats. However, Ethan does not want to sit beside Dianne. In how many ways can they arrange themselves in the row of seats? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
When the circumference of a toy balloon is increased from 20 inches to 25 inches, what is the increase in radius? Express your answer in inches and include the pi symbol if necessary. | \dfrac{5}{2\pi} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A complex number $z$ satisfies $|z|=|z-1|=|z-i|$. Find the value of $|z|$. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Four students go to a shop to buy either a brush pen or a pencil. Each student buys exactly one item, and at least one person buys a pencil. Calculate the total number of different buying combinations possible. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are not collinear, and $2\overrightarrow{OP}=x\overrightarrow{OA}+y\overrightarrow{OB}$. If $\overrightarrow{PA}=λ\overrightarrow{AB}$ ($λ∈R$), derive the relationship satisfied by $x$ and $y$. Express your answer as an equation. | x + y - 2 = 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the vertex coordinates of the parabola y = (2x + 1)^2 - 3. Express your answer as an ordered pair (x, y). | \left(-\frac{1}{2}, -3\right) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \sqrt {1-x^{2}}$, calculate the value of $\int_{-1}^{1} f(x)dx$. Express your answer as a fraction of pi, if applicable. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number. Express your answer as a single integer. | 3785 | big_math | [
"Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $AB=\sqrt{5}$, $AC=\sqrt{3}$, $BC=2\sqrt{2}$. Find the measure of angle A in degrees. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z satisfies z(1+i) = 2i, calculate the value of z. Express your answer as a complex number in the form a+bi. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function with the domain of R, and f(1+x) = f(-x). If f(-1/3) = 1/3, calculate the value of f(5/3). Express your answer as a single fraction or decimal value. | 1/3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given circles M: x^{2}+(y+1)^{2}=1 and N: (x-2)^{2}+(y-3)^{2}=1 are symmetric with respect to the line l, find the equation of l. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x + 2y - 3 = 0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|=2$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, calculate the dot product $\overrightarrow{a}\cdot \overrightarrow{b}$. Provide your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
There are four statements on a card, and only these four:
There is exactly one false statement on this card.
There are exactly two false statements on this card.
There are exactly three false statements on this card.
There are exactly four false statements on this card.
(Assume each statement on the card is either true... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest integer n that satisfies the inequality n^{200} < 5^{300}. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
To send an email, the following steps are involved: a. Open the email box; b. Enter the recipient's address; c. Enter the subject; d. Enter the content of the email; e. Click "Compose"; f. Click "Send". Determine the correct sequence of these steps, separating each step with an arrow (→). | a→e→b→c→d→f | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of different arrangements for 7 people to stand in a row where person A cannot stand at the head and person B cannot stand in the middle. Provide your answer as a single integer. | 3720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The bookshelf in the library has three tiers. The first tier contains 3 different mathematics books, the second tier has 5 different language books, and the third tier has 8 different English books. Calculate the total number of different ways to select one book from the bookshelf. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A complex number z satisfies the equation (z-i)(2-i) = i. Find the magnitude |z|. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
A store had a sale on T-shirts. For every two T-shirts purchased at the regular price, a third T-shirt was bought for $1.00. Twelve T-shirts were bought for $120.00. Find the regular price of one T-shirt. Express your answer in dollars and cents. | 14.50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a} =(2,λ)$ and $\vec{b} =(-4,10)$, and $\vec{a} \perp \vec{b}$, calculate the value of the real number λ using the dot product property of perpendicular vectors. | \frac{4}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Four different balls are to be placed into three boxes labeled 1 to 3, with no box left empty. Calculate the number of ways to do this. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children is 42. Calculate the sum of the ages of the oldest three children. | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(c_{20}^{2n+6}=c_{20}^{n+2}\) (\(n\in \mathbb{N}\)), and \((2-x)^n=a_0+a_1x+a_2x^2+\cdots+a_nx^n\), calculate the value of \(a_0-a_1+a_2-\ldots+(-1)^na_n\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A fraction with the numerator 1 and a denominator greater than 1 is called a unit fraction. If 1/8 is represented as the sum of two different unit fractions, how many possible representations are there? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the circle has centre O. The shaded sector AOB has a sector angle of 90 degrees, and AB has an arc length of 2π units. Calculate the area of sector AOB. Express your answer in terms of π. | 4\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $M=\{x|0\leqslant x\leqslant 1\}$, $N=\{x|x^{2}\geqslant 1\}$. Find the result of the union of set M and the complement of set N. Express your answer in interval notation. | (-1,1] | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A computer user plans to spend no more than 500 yuan to purchase software and boxed disks, with unit prices of 60 yuan and 70 yuan, respectively. According to the requirements, at least 3 pieces of software and at least 2 boxes of disks must be purchased. Calculate the number of different purchasing methods that satisf... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In a Cartesian coordinate system, find the angle of inclination in degrees for the line $x + \sqrt{3}y + 1 = 0$. Provide your answer as a numerical value without the degree symbol. | 150 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all zeros of the function $f(x)=\begin{cases} & {(\frac{1}{2})}^{x}-2, & x < 0, \\ & x-1, & x\geqslant 0 \\ \end{cases}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the complex number z in the equation (z - i)(2 - i) = 5. Express your answer in the form a + bi, where a and b are real numbers. | 2 + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \(\sqrt{36 \times \sqrt{16}}\) and provide the result as a whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
If the third term of the expansion of \\((x-a)^{6}\\) is \\(60x^{4}\\), then find the value of \\(a\\). Express your answer as a numerical value, including the ± symbol if applicable. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system, find the coordinates of the point that is symmetric to point $P(3,1,5)$ about the origin. Express your answer as an ordered triple $(x, y, z)$, including negative signs if applicable. | (-3,-1,-5) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 weights of 1 gram, 3 weights of 2 grams, and 1 weight of 5 grams. Weights can only be added to one side of the scale. How many different ways are there to weigh an object of 9 grams? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest number in the set {3.2, 2.3, 3, 2.23, 3.22}. Express your answer as a decimal number, including all necessary decimal places. | 2.23 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that vector $\overrightarrow{a}=\left(x,2\right)$, $\overrightarrow{b}=\left(2,3\right)$, $\overrightarrow{c}=\left(2,-4\right)$, and $\overrightarrow{a} \parallel \overrightarrow{c}$, calculate the projection vector of $\overrightarrow{a}$ onto $\overrightarrow{b}$. Express your answer as a vector in the format ... | \left(\frac{8}{13}, \frac{12}{13}\right) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the value of the complex number $\frac{1+3i}{1-i}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
For the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$, find the length of a chord that is drawn through one of the foci and is perpendicular to the major axis. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{-1, 1\}$ and $B=\{0, 2\}$, find the number of elements in the set $\{z|z=x+y, x\in A, y\in B\}$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \frac{\cos x}{e^x}$, find the equation of the tangent line to the graph of $f(x)$ at the point $(0, f(0))$. Express your answer in the standard form $Ax + By + C = 0$. | x + y - 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
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