ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? The original answer is in the form of \(\frac{k}{m}\) where the fraction is in simplest form. Please find the value of k + m.
Answer: 71
Problem: A sequence \((a_1,b_1), (a_2,b_2), (a_3,b_3), \ldots\) of points in the coordinate plane satisfies
\((a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)\) for \(n = 1,2,3,\cdots\).
Suppose that \((a_{100},b_{100}) = (2,4)\). Find the value of \(a_1 + b_1\). The original answer is in the form \(\frac{x}{y \cdot 2^{z}}\), where y is not divisible by 2. Please provide the value of \(x + y + z\).
Answer: 100
Problem: Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. If the area of hexagon $ABQCDP$ is $x\sqrt{3}$, what is the value of $x$?
Answer: 30
Problem: The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? Provide your answer as an integer.
Answer: 6
Problem: A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths \(15\) and \(25\) meters. What fraction of the yard is occupied by the flower beds? The answer should be in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in its simplest form. Please find the value of \(k + m\).
Answer: 6
Problem: A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds? The original answer is in the form \frac{k}{m}, where k and m are integers, and the fraction is in its simplest form. Please give the value of k + m.
Answer: 6
Problem: What is the sum of the digits of the square of $111111111$? Provide your answer as a single integer.
Answer: 81
Problem: Kiana has two older twin brothers. The product of their three ages is $128$. What is the sum of their three ages?
Answer: 18
Problem: By inserting parentheses, it is possible to give the expression
\[2\times3 + 4\times5\]
several values. How many different values can be obtained?
Answer: 4
Problem: A carton contains milk that is \(2\%\) fat, an amount that is \(40\%\) less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Provide your answer as a fraction in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
Answer: 13
Problem: Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
Answer: 3
Problem: Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the value of $\angle D$ in degrees? If the answer is $x^\circ$, what is the value of $10x$?
Answer: 525
Problem: Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have? Fill in the blank with the number of sides.
Answer: 23
Problem: A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. If the answer is $x$ meters, what is the value of $10x$?
Answer: 24
Problem: A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 3
Problem: On a checkerboard composed of \(64\) unit squares, find the probability that a randomly chosen unit square does not touch the outer edge of the board. Express your answer in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of \(k + m\).
Answer: 25
Problem: How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?
Answer: 6
Problem: The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?
Answer: 761
Problem: The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil?
Answer: 4
Problem: Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
Answer: 3
Problem: On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? If Monday is day 1, what is the number of the day?
Answer: 5
Problem: Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. The original answer is in $\frac{k}{m}$ format, where $\frac{k}{m}$ is in simplest form. Please find the value of $k + m$.
Answer: 5
Problem: For what value of $n$ is $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$? Find the value of $n$.
Answer: 97
Problem: A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make? Express your answer in the form of an improper fraction \frac{k}{m}, and find the value of k + m.
Answer: 39
Problem: Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. Find the sum of all possible values of $|a-d|$.
Answer: 18
Problem: Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects \(\overline{BO}\) at \(D\). The original answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 3
Problem: Trapezoid \(ABCD\) has \(AD \parallel BC\), \(BD = 1\), \(\angle DBA = 23^\circ\), and \(\angle BDC = 46^\circ\). The ratio \(BC: AD\) is \(9: 5\). What is the value of \(CD\)? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a fully simplified fraction. Please provide the value of \(k + m\).
Answer: 9