MathDB

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

n

problems and get them all correct, you will receive

\max(0, (n-1)(n-2))

points. If any of them are wrong (or you leave them all blank), you will receive

0

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

12

points if all five answers are correct, 0 points if any are wrong).
(a) Does

\sum_{i=1}^{p-1}\frac{1}{i}\equiv 0\pmod{p^2}

for all odd prime numbers

p

? (Note that

\frac{1}{i}

denotes the number such that

i\cdot\frac{1}{i}\equiv 1\pmod{p^2}

)
(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 [color = red]different 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\dots

a rational number?
(f) Do there exist 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

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.

Problem Statement