HMMT 二月 2017 · 冲刺赛 · 第 36 题

HMMT February 2017 — Guts Round — Problem 36

[ 20 ] ∑ p − 1 1 2 1 (a) Does ≡ 0 (mod p ) for all odd prime numbers p ? (Note that denotes the number such i =1 i i 1 2 that i · ≡ 1 (mod p )) i (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 × 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? ∏ 2 2 2 2 2 p +1 2 +1 3 +1 5 +1 7 +1 (e) Is = · · · · . . . a rational number? 2 2 2 2 2 p p − 1 2 − 1 3 − 1 5 − 1 7 − 1 (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?

解析

[ 20 ] ∑ p − 1 1 1 2 (a) Does ≡ 0 (mod p ) for all odd prime numbers p ? (Note that denotes the number such i =1 i i 1 2 that i · ≡ 1 (mod p )) i (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 × 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? ∏ 2 2 2 2 2 p +1 2 +1 3 +1 5 +1 7 +1 (e) Is = · · · · . . . a rational number? 2 2 2 2 2 p p − 1 2 − 1 3 − 1 5 − 1 7 − 1 (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? Proposed by: Alexander Katz Answer: NYYYYY