HMMT February

(a) Does $\sum_{i=1}^{p-1} \frac{1}{i} \equiv 0\left(\bmod p^{2}\right)$ for all odd prime numbers $p$? (Note that $\frac{1}{i}$ denotes the number such that $\left.i \cdot \frac{1}{i} \equiv 1\left(\bmod p^{2}\right)\right)$ (b) Do there exist 2017 positive perfect cubes that sum to a perfect cube? (c) Does there exist a right triangle with rational side lengths and area 5? (d) A magic square is a $3 \times 3$ grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all prime numbers? (e) Is $\prod_{p} \frac{p^{2}+1}{p^{2}-1}=\frac{2^{2}+1}{2^{2}-1} \cdot \frac{3^{2}+1}{3^{2}-1} \cdot \frac{5^{2}+1}{5^{2}-1} \cdot \frac{7^{2}+1}{7^{2}-1} \cdot \ldots$ a rational number? (f) Do there exist an infinite number of pairs of distinct integers $(a, b)$ such that $a$ and $b$ have the same set of prime divisors, and $a+1$ and $b+1$ also have the same set of prime divisors?