MathDB

36

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(1)

2017 Guts #36: USAYNO Number Theory

Source:

2/21/2017
Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer nn problems and get them all correct, you will receive max⁡(0,(n−1)(n−2))\max(0, (n-1)(n-2)) points. If any of them are wrong (or you leave them all blank), you will receive 00 points.
Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive 1212 points if all five answers are correct, 0 points if any are wrong).
(a) Does ∑i=1p−11i≡0(modp2)\sum_{i=1}^{p-1}\frac{1}{i}\equiv 0\pmod{p^2} for all odd prime numbers pp? (Note that 1i\frac{1}{i} denotes the number such that i⋅1i≡1(modp2)i\cdot\frac{1}{i}\equiv 1\pmod{p^2})
(b) Do there exist 20172017 positive perfect cubes that sum to a perfect cube?
(c) Does there exist a right triangle with rational side lengths and area 55?
(d) A magic square is a 3×33\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 [color = red]different prime numbers?
(e) Is ∏pp2+1p2−1=22+122−1⋅32+132−1⋅52+152−1⋅72+172−1⋅…\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\dots a rational number?
(f) Do there exist infinite number of pairs of distinct integers (a,b)(a,b) such that aa and bb have the same set of prime divisors, and a+1a+1 and b+1b+1 have the same set of prime divisors?
[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.
[color = red]A clarification was issued for problem 36(d) during the test. I have included it above.
USAYNO