rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: For positive integers  $n,$  let  $\nu_3 (n)$  denote the largest integer  $k$  such that  $3^k$  divides  $n.$  Find the number of subsets  $S$  (possibly containing 0 or 1 elements) of  $\{1, 2, \ldots, 81\}$  such that for any distinct  $a,b \in S$ ,  $\nu_3 (a-b)$  is even.

*Author: Alex Zhu*

<details><summary>Clarification</summary>We only need  $\nu_3(a-b)$  to be even for  $a>b$ .</details>
Answer: 2^{81}
Problem:  $11$  theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least  $1$  performance of every other group. At least how many days did the festival last?
Answer: 6
Problem: Find all positive integers that can be represented as   $$ \frac{abc+ab+a}{abc+bc+c} $$  for some positive integers  $a, b, c$ .
Answer: 1 \text{ and } 2
Problem: Ket  $f(x) = x^{2} +ax + b$ . If for all nonzero real  $x$   $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$  and the roots of  $f(x) = 0$  are integers, what is the value of  $a^{2}+b^{2}$ ?
Answer: 13
Problem: Find all the functions  $ f: \mathbb{N}\rightarrow \mathbb{N}$  such that
\[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]
Answer:  f(n) = n
Problem: Determine all squarefree positive integers  $n\geq 2$  such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where  $d_1,d_2,\ldots,d_k$  are all the positive divisors of  $n$ .
Answer: n = 6