ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Frank Narf accidentally read a degree $n$ polynomial with integer coefficients backwards. That is, he read $a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ as $a_{0} x^{n}+\ldots+a_{n-1} x+a_{n}$. Luckily, the reversed polynomial had the same zeros as the original polynomial. All the reversed polynomial's zeros were real, and also integers. If $1 \leq n \leq 7$, compute the number of such polynomials such that $\operatorname{GCD}\left(a_{0}, a_{1}, \ldots, a_{n}\right)=1$.
Answer: 70
Problem: Given a regular 16-gon, extend three of its sides to form a triangle none of whose vertices lie on the 16-gon itself. Compute the number of noncongruent triangles that can be formed in this manner.
Answer: 11
Problem: Two square tiles of area 9 are placed with one directly on top of the other. The top tile is then rotated about its center by an acute angle $\theta$. If the area of the overlapping region is 8 , compute $\sin \theta+\cos \theta$.
Answer: \frac{5}{4}
Problem: Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$.
Answer: 12
Problem: Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$.
Answer: 150
Problem: Suppose that $r$ and $s$ are the two roots of the equation $F_{k} x^{2}+F_{k+1} x+F_{k+2}=0$, where $F_{n}$ denotes the $n^{\text {th }}$ Fibonacci number. Compute the value of $(r+1)(s+1)$.
Answer: 2
Problem: Let $T=2$. Compute the product of $-T-i$ and $i-T$, where $i=\sqrt{-1}$.
Answer: 5
Problem: Let $T=5$. Compute the number of positive divisors of the number $20^{4} \cdot 11^{T}$ that are perfect cubes.
Answer: 12
Problem: Let $T=72 \sqrt{2}$, and let $K=\left(\frac{T}{12}\right)^{2}$. In the sequence $0.5,1,-1.5,2,2.5,-3, \ldots$, every third term is negative, and the absolute values of the terms form an arithmetic sequence. Compute the sum of the first $K$ terms of this sequence.
Answer: 414
Problem: Let $A$ be the sum of the digits of the number you will receive from position 7 , and let $B$ be the sum of the digits of the number you will receive from position 9 . Let $(x, y)$ be a point randomly selected from the interior of the triangle whose consecutive vertices are $(1,1),(B, 7)$ and $(17,1)$. Compute the probability that $x>A-1$.
Answer: \frac{79}{128}
Problem: Let $T=9.5$. If $\log _{2} x^{T}-\log _{4} x=\log _{8} x^{k}$ is an identity for all $x>0$, compute the value of $k$.
Answer: 27
Problem: Let $T=16$. An isosceles trapezoid has an area of $T+1$, a height of 2 , and the shorter base is 3 units shorter than the longer base. Compute the sum of the length of the shorter base and the length of one of the congruent sides.
Answer: 9.5
Problem: Let $T=10$. Susan flips a fair coin $T$ times. Leo has an unfair coin such that the probability of flipping heads is $\frac{1}{3}$. Leo gets to flip his coin the least number of times so that Leo's expected number of heads will exceed Susan's expected number of heads. Compute the number of times Leo gets to flip his coin.
Answer: 16
Problem: Let $T=1$. Dennis and Edward each take 48 minutes to mow a lawn, and Shawn takes 24 minutes to mow a lawn. Working together, how many lawns can Dennis, Edward, and Shawn mow in $2 \cdot T$ hours? (For the purposes of this problem, you may assume that after they complete mowing a lawn, they immediately start mowing the next lawn.)
Answer: 10
Problem: Let T be a rational number. Compute $\sin ^{2} \frac{T \pi}{2}+\sin ^{2} \frac{(5-T) \pi}{2}$.
Answer: 1
Problem: Let $T=11$. Compute the value of $x$ that satisfies $\sqrt{20+\sqrt{T+x}}=5$.
Answer: 14
Problem: The sum of the interior angles of an $n$-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. Compute $n$.
Answer: 11