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 |
|---|---|---|---|---|---|
A belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, calculate the distance between the centers of the two pulleys. Express your answer in inches. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
From the set of natural numbers {1,2,3,4,5}, randomly select 3 distinct numbers. Calculate the probability that the average of these 3 numbers is greater than 3. Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are positive numbers, and 1/x + 4/y = 1, find the minimum value of x+y. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $z=(-1+i)^2$. Express your answer as a numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle \\(\\alpha\\) passes through the point \\(P(m,-3)\\), and \\(\\cos \\alpha = -\\frac{4}{5}\\), find the value of \\(m\\). Express your answer as a numerical value. | -4 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that 110n^3 has 110 distinct positive divisors, including 1 and 110n^3 itself, and that the prime factorization of 110 is 2 * 5 * 11, determine the number of distinct positive divisors of 81n^4. Express your answer as a single integer. | 325 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0 | 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"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 3^a + log_3 a = 3^(2b) + log_3 b, determine the relationship between a and 2b. Express your answer as an inequality. | a < 2b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle (x-2)^2 + y^2 = 4 and the line of symmetry y = sqrt(3)/3 * x, find the equation of the circle symmetric to the given circle with respect to the line of symmetry. Express your answer in standard form. | (x-1)^2 + (y-sqrt(3))^2 = 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f: x ↦ x^2 is a mapping from set A to set B = {0, 1, 4}, calculate the maximum number of elements in set A. | 5 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a*sin(x) + b*x^3 + 4, where f'(x) is the derivative of f(x), find the value of f(2014) + f(-2014) + f'(2015) - f'(-2015). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
A convex polygon has interior angles that are in arithmetic progression. The common difference is 5 degrees, and the largest angle is 160 degrees. Find the number of sides of the polygon. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Three circles with radii 2, 3, and 3 touch each other. Find the area of the triangle formed by joining the centers of these circles. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the tangent line at point (1, -2) on the curve y = ln(x) - 2x and the coordinate axes. Express your answer as a decimal or fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.421875 | 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"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the high school mathematics exam scores, denoted by ξ, of a certain school approximately follow a normal distribution N(100, 5^2) and P(ξ < 110) = 0.98, calculate the value of P(90 < ξ < 100). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.48 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies the equation \\(\\dfrac{{\\bar{z}}}{1-i}=i\\), where \\(\\bar{z}\\) is the conjugate of the complex number z, calculate the modulus of z, denoted as |z|. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Scott challenges Chris to a 100-meter race. Scott runs 4 meters for every 5 meters that Chris runs. How far will Scott have run when Chris crosses the finish line? Express your answer in meters as a whole number. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, if $S_3=9$, $S_6=36$, calculate the value of $S_9$. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $z^x = y^{2x}$, $2^z = 2\\cdot4^x$, and $x + y + z = 16$ to find the integral roots in the order $x, y, z$. Express your answer as a list of three integers. | 4, 3, 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number 0.000016 in scientific notation. Provide your answer in the format $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 1.6 \times 10^{-5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations \(x(t) = 3\cos(t)\) and \(y(t) = 4\sin(t)\), find the maximum distance between the particle and the origin. Express your answer in meters with two decimal places. | 4.00 \text{ m} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all positive odd numbers less than 20. Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola $x^{2}=4y$. Express your answer as a coordinate pair (x, y). | (0,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
If $\alpha$ is in the third quadrant, simplify the expression $$\frac {\cos\alpha}{ \sqrt {1-\sin^{2}\alpha}}+ \frac {2\sin\alpha}{ \sqrt {1-\cos^{2}\alpha}}.$$ Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\log_2(a) + \log_2(b) \geq 6$, find the least possible value of $a + b$. Provide your answer as a single number. | 16 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ where $a_1=1$, $a_2=2$, and $a_{n+2}=a_{n+1}+a_n$, calculate the value of $a_5$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A survey of 200 students was conducted to determine their favorite cafeteria food. The results showed that 25% of students chose cookies, 30% chose pizza, and 35% chose pasta. Calculate the number of students who chose sandwiches as their favorite food. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with the sum of its first $n$ terms denoted as $S_n$, where $a_1=2$ and $S_4=20$, calculate the sum of its first 6 terms, $S_6$. Express your answer as a single numerical value. | 42 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The area of a sector is $6\,cm^2$, and its radius is $2\,cm$. Find the central angle of the sector. Express your answer as a numerical value without units. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, find the coordinates of the point corresponding to the conjugate of the complex number $\frac{2-i}{1+i}$. Express your answer as a coordinate pair (x, y) and clearly indicate the quadrant in which the point lies. | \left(\frac{1}{2}, \frac{3}{2}\right) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 21 into binary. Express your answer in binary format with a subscript 2 to denote the base (e.g., 10101_{(2)}). | 10101_{(2)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $x^{2}=2py$ with its focus at $F(0,2)$, find the value of $p$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the point M(x,y) moves on the line x+2y+1=0, find the minimum value of 2^x + 4^y. | \sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = -2x + m, find the value of m that makes this function a direct proportion with respect to x. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 different books placed in a row on a desk, with the requirement that books A and B must be placed at the two ends, and books C and D must be adjacent. Find the number of different arrangements of the books. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\frac {a+2i}{i}=b+i$$ (where $a$, $b$ are real numbers, and $i$ is the imaginary unit), calculate the value of $a+b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
The cost of 1 piece of gum is 1 cent. Calculate the total cost of 1000 pieces of gum in dollars, rounded to two decimal places. | 10.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
It's known that {a<sub>n</sub>} is an arithmetic sequence, with a<sub>2</sub> + a<sub>8</sub> = 12. Find the value of a<sub>5</sub>. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest integer \( n \) for which \( 3\left(n^{2007}\right) < 3^{4015} \)? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = x^2 - 2x + 2, where x ∈ [-3, 2], find the range of this function. Express your answer as an interval in standard notation (e.g., [a, b]). | [1, 17] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the "Home Appliances to the Countryside" campaign, a factory plans to select 3 people from a group consisting of 5 salespersons and 4 technicians to go to a nearby town for a home appliance promotion. If it is required that at least one salesperson and one technician are included, calculate the number of different c... | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given a mapping \( f \) from the set \( M = \{a, b, c\} \) to the set \( N = \{-3, -2, \cdots, 3\} \), determine the number of mappings \( f \) that satisfy the condition \( f(a) + f(b) + f(c) = 0 \). Express your answer as a whole number. | 37 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( m + 1 = \frac{n - 2}{3} \), find the value of \( 3m - n \). Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The annual incomes of 1,000 families range from $8,200 to $98,000. In error, the largest income was entered on the computer as $980,000 instead of $98,000. Calculate the difference between the mean of the incorrect data and the mean of the actual data. Express your answer in dollars as a whole number. | 882 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of a set of six numbers is 10. If the number 25 is removed from the set, calculate the mean of the remaining numbers. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overset{⇀}{a}$ and $\overset{⇀}{b}$ on a plane, $|\overset{⇀}{a}| = 2$ and $\overset{⇀}{a} \perp (\overset{⇀}{a} + 2\overset{⇀}{b})$, determine the projection of vector $\overset{⇀}{b}$ on vector $\overset{⇀}{a}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n} a_{n}=2n-1$. Calculate the sum of the first 60 terms of the sequence. Express your answer as a single integer. | 1830 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function $f(x)=\frac{\sqrt{x-2}}{x-3}+\log_{10}(4-x)$. Express your answer in interval notation, including any necessary union or intersection symbols. | [2,3) \cup (3,4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \( 3 \times (7 - 5) - 5 \) and provide your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \(\frac{1 + 0.25}{2 \times \frac{3}{4} - 0.75} + \frac{3 \times 0.5}{1 \frac{1}{2} + 3}\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of (2 * sin^2(25°) - 1) / (sin(20°) * cos(20°)). Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Factor the quadratic equation $x^{2}+4x-1=0$ using the completing the square method. Express your answer in the form $(x+a)^{2}=b$. | $(x+2)^{2}=5$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases} 2^{x}, & x > 0 \\ x+1, & x\leqslant 0 \end{cases}$, find $f(-2)$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $3+4i$. Express your answer as a complex number in the form $a+bi$. | 3-4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the right focus of the hyperbola x^2/m - y^2/5 = 1 coincides with the focus of the parabola y^2 = 12x, determine the equation of the asymptotes of this hyperbola. Express your answer in the form y = ±kx, where k is a constant. | y = \pm \frac{\sqrt{5}}{2}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A certain 3D object has an edge with length 2. In the front view of this object, the projection of this edge is a line segment with length 2. In the side view and top view of the object, the projections of this edge are line segments with lengths a and b, respectively. Calculate the maximum possible value of a + b. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real numbers $a$ and $b$ satisfy $ab > 0$, find the minimum value of the expression $a^2 + 4b^2 + \frac{1}{ab}$. Provide your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations $\left\{\begin{array}{c}x=m\\ y=n\end{array}\right.$ and the equation $x-y=2$, find the algebraic expression $n-m$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, and $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, calculate the magnitude of the vector $|2\overrightarrow{a} - \overrightarrow{b}|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, a point is randomly chosen within the square formed by the lines $x=1$, $y=1$ and the coordinate axes. Calculate the probability that this point falls within the region below the curve $y=x^{2}$. Express your answer as a fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x / (x + 2) at the point (-1, -1). Express your answer in slope-intercept form (y = mx + b). | y = 2x + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select two numbers x and y in the interval [0,1]. Calculate the probability that y > 3x. Express your answer as a simplified fraction. | \dfrac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A small ink cartridge has enough ink to print 600 pages. Three small cartridges can print as many pages as two medium cartridges. Three medium cartridges can print as many pages as two large cartridges. Calculate the number of pages that can be printed using a large cartridge. | 1350 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A child has a set of 96 different blocks. Each block can be either plastic or wood, come in three sizes (large, medium, small), be one of four colors (blue, green, red, yellow), and have one of four shapes (circle, hexagon, square, triangle). Find the number of blocks that differ in exactly two attributes from a block ... | 29 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle P Q R$ is equilateral and has a side length of $6 \mathrm{~cm}$. Each of the shaded triangles is equilateral and has a side length of $2 \mathrm{~cm}$. Calculate the fraction of the area of $\triangle P Q R$ that is shaded. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given $3\\vec{a} + 4\\vec{b} + 5\\vec{c} = 0$ and $|\\vec{a}| = |\\vec{b}| = |\\vec{c}| = 1$, calculate the value of $\\vec{b} \\cdot (\\vec{a} + \\vec{c})$. Express your answer as a single fraction, including the negative sign if applicable. | -\\dfrac{4}{5} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of cell divides once every 30 minutes, with one cell dividing into two. Calculate the number of cells that can be obtained from one such cell after 4 hours and 30 minutes. Express your answer as a whole number. | 512 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains. Calculate the number of tetrads that can be produced by this cell during meiosis, given that each tetrad is formed by two homologous chromosomes, each having two sister chromatids. | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, point \( P \) is on the number line at 3 and \( V \) is at 33. The number line between 3 and 33 is divided into six equal parts by the points \( Q, R, S, T, U \). Find the sum of the lengths of \( P S \) and \( T V \). Express your answer as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the point P(a,a) (a ≠ 0) lies on the terminal side of angle α, calculate the value of sin(α). Express your answer as a single fraction or a range of fractions, including any necessary absolute value signs or ± symbols. | \frac{\sqrt{2}}{2} \text{ or } -\frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Nine balls numbered 1, 2, …, 9 are placed into a bag. Person A draws a ball from the bag, and its number is a. After putting it back, person B draws another ball from the bag, and its number is b. Calculate the probability that the event a - 2b + 10 > 0 occurs. Express your answer as a fraction in simplest form. | \frac{61}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Two vehicles, Vehicle A and Vehicle B, shuttle back and forth between two locations that are 12 kilometers apart. Vehicle A stops every 0.5 kilometers, while Vehicle B stops every 0.8 kilometers. Find the shortest distance between two stations where Vehicles A and B do not stop at the same place. Express your answer in... | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)= \begin{cases} \left( \frac {1}{2}\right)^{x}-7 & (x < 0) \\ \sqrt {x} & (x\geqslant 0)\end{cases}$. Find the range of the real number $a$ such that $f(a) < 1$. Express your answer in interval notation. | (-3, 1) | big_math | [
"Mathematics -> Functions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by $a_n = \dfrac{1}{n(n+1)}$, calculate the sum of the first 5 terms, $S_5$. Express your answer as a fraction in simplest form. | \dfrac{5}{6} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given two quadratic equations:
1. $(1992x)^2 - 1991 \cdot 1993 x - 1 = 0$ with the larger root denoted as $m$.
2. $x^2 + 1991x - 1992 = 0$ with the smaller root denoted as $n$.
Find the difference $m - n$. Express your answer as a single numerical value. | 1993 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\sqrt {3}, 3, \sqrt {15}, \sqrt {21}, 3\sqrt {3}, \ldots$, find the term number of the number 9. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the measure of an angle α is 13°14', solve for x in the equation 2α - x = 180° - 3x. Express your answer in degrees and minutes (° '). | 76°46' | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Quadrilateral ABCD is a parallelogram in the complex plane. The complex numbers corresponding to points A, B, and C are 1+3i, 2-i, and -3+i, respectively. Calculate the complex number corresponding to point D. Express your answer in the form a + bi. | -4+5i | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\left\{a_n\right\}$, where $a_5=13$ and $S_5=35$, find the common difference $d$. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, find the equation of the line containing the chord with midpoint $(2, \frac{3}{2})$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 3x + 4y - 12 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given lines $l_1$: $x+2y+t^{2}=0$ and $l_2$: $2x+4y+2t-3=0$, determine the value of $t$ that minimizes the distance between $l_1$ and $l_2$. Express your answer as a single value of $t$. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the order of the three numbers from smallest to largest: $a=\left( \frac{3}{4} \right)^{- \frac{1}{3}}$, $b=\left( \frac{3}{4} \right)^{- \frac{1}{4}}$, and $c=\left( \frac{3}{2} \right)^{- \frac{1}{4}}$. Express your answer as an ordering of the three variables (e.g., a < b < c). | c < b < a | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the eccentricity of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. Express your answer as a fraction with a square root in the numerator. | \dfrac{\sqrt{7}}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of b such that the line y = (1/2)x + b is tangent to the curve y = (-1/2)x + ln(x). Express your answer as a single number. | b=-1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, where a_2 + a_4 = 16 and a_1 = 1, find the value of a_5. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
When two dice are thrown consecutively, the numbers obtained are m and n. Calculate the probability that point P(m, n) lies on the line x+y=4. Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence $\{a_{n}\}$, given $a_{2}=4$, $a_{4}=16$, and $a_{1}a_{2}a_{3}\ldots \ldots a_{m}=a^{2}_{m+1}$, calculate the value of $m$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $2000(2000^{2000})$. Express your answer as a simplified exponential expression. | 2000^{2001} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of bacteria divides once every half hour (one splits into two). If there is initially 1 bacterium, calculate the total number of bacteria after 3 hours. Express your answer as a single numerical value. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle, the acute angles are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. Find the smallest possible value of $b$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.555556 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of the integer 100. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given a curve y = x^2 / 4, find the x-coordinate of the tangent point where the slope of the tangent line is 1/2. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A right-angled triangle has a sum of the lengths of its three sides equal to 16 cm. The sum of the squares of the lengths of the three sides of the triangle is 98 cm^2. Find the area of the triangle. Express your answer in square centimeters (cm^2) as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given $z \in \mathbb{C}$, and $|z + 2 - 2i| = 1$, calculate the minimum value of $|z - 2 - 2i|$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + bx + c, and f(1) = 0, f(3) = 0, find the value of f(-1). Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The numbers 1 to 8 are to be placed, one per circle, in the circles shown. The number next to each arrow shows what the product of the numbers in the circles on that straight line should be. Calculate the sum of the numbers in the three circles at the bottom of the diagram. Use each number from 1 to 8 exactly once. | 17 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. Calculate the cars' combined rate of miles per gallon of gasoline. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{(3!)!}{3!}$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
What is the absolute value of -6? Provide your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
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