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HMMT 二月 2018 · 冲刺赛 · 第 7 题

HMMT February 2018 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] A triple of integers ( a, b, c ) satisfies a + bc = 2017 and b + ca = 8. Find all possible values of c .
解析
  1. [ 6 ] A triple of integers ( a, b, c ) satisfies a + bc = 2017 and b + ca = 8. Find all possible values of c . Proposed by: Ashwin Sah Answer: − 6 , 0 , 2 , 8 Add and subtract the two equations to find ( b + a )( c + 1) = 8 + 2017 , ( b − a )( c − 1) = 2017 − 8 . We see that c is even and then that every integer c with c + 1 | 2025 , c − 1 | 2009 works. We factor and solve. The full solutions are (2017 , 8 , 0) , ( − 667 , 1342 , 2) , ( − 59 , − 346 , − 6) , ( − 31 , 256 , 8).