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HMMT 二月 2003 · 冲刺赛 · 第 26 题

HMMT February 2003 — Guts Round — Problem 26

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [9] Find all integers x such that x + 6 x + 28 is a perfect square.
解析

英文解析

  1. Find all integers m such that m + 6 m + 28 is a perfect square.
    Solution: 6 , − 12
    2 2 2
    We must have m +6 m +28 = n , where n is an integer. Rewrite this as ( m +3) +19 =
    2 2 2
    n ⇒ n − ( m + 3) = 19 ⇒ ( n − m − 3)( n + m + 3) = 19. Let a = n − m − 3 andb = n + m + 3, so we want ab = 19. This leaves only 4 cases:
    • a = 1, b = 19. Solve the system n − m − 3 = 1 and n + m + 3 = 19 to get n = 10
    and m = 6, giving one possible solution.
    • a = 19, b = 1. Solve the system, as above, to get n = 10 and m = − 12.
    • a = − 1, b = − 19. We get n = − 10 and m = − 12.
    • a = − 19, b = − 1. We get n = − 10 and m = 6.
    Thus the only m are 6 and − 12.