HMMT 十一月 2015 · 冲刺赛 · 第 11 题

HMMT November 2015 — Guts Round — Problem 11

[ 8 ] Find all integers n , not necessarily positive, for which there exist positive integers a, b, c satisfying n n n a + b = c .

解析

[ 8 ] Find all integers n , not necessarily positive, for which there exist positive integers a, b, c satisfying n n n a + b = c . Proposed by: Rikhav Shah Answer: ± 1 , ± 2 n n n − n By Fermat’s Last Theorem, we know n < 3. Suppose n ≤ − 3. Then a + b = c = ⇒ ( bc ) + − n − n ( ac ) = ( ab ) , but since − n ≥ 3, this is also impossible by Fermat’s Last Theorem. As a result, | n | < 3. 0 0 0 Furthermore, n 6 = 0, as a + b = c = ⇒ 1 + 1 = 1, which is false. We now just need to find constructions for n = − 2 , − 1 , 1 , 2. When n = 1, ( a, b, c ) = (1 , 2 , 3) suffices, and when n = 2, ( a, b, c ) = (3 , 4 , 5) works nicely. When n = − 1, ( a, b, c ) = (6 , 3 , 2) works, and when n = − 2, ( a, b, c ) = (20 , 15 , 12) is one example. Therefore, the working values are n = ± 1 , ± 2 .