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HMMT 二月 2009 · 冲刺赛 · 第 18 题

HMMT February 2009 — Guts Round — Problem 18

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 9 ] Find the positive integer n such that n + 2 n + 9 n + 8 is the cube of an integer.
    12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth

英文原题

[ 9 ] Find the positive integer n such that n 3 + 2 n 2 + 9 n + 8 is the cube of an integer.
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12 th HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUND

解析

英文解析

  1. [ 9 ] If n is a positive integer such that n + 2 n + 9 n + 8 is the cube of an integer, find n .
    Answer: 7
    3 3 2 3 3 2 3
    Solution: Since n < n + 2 n + 9 n + 8 < ( n + 2) , we must have n + 2 n + 9 n + 8 = ( n + 1) .
    Thus n = 6 n + 7, so n = 7.2
    12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth