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HMMT 十一月 2013 · 冲刺赛 · 第 30 题

HMMT November 2013 — Guts Round — Problem 30

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Find the number of ordered triples of integers ( a, b, c ) with 1 ≤ a, b, c ≤ 100 and a b + b c + c a =
    2 2 2
    ab + bc + ca .
    HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND

英文原题

[ 15 ] Find the number of ordered triples of integers ( a, b, c ) with 1 ≤ a, b, c ≤ 100 and a 2 b + b 2 c + c 2 a =
ab 2 + bc 2 + ca 2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND

解析

英文解析

  1. [ 15 ] Find the number of ordered triples of integers ( a, b, c ) with 1 ≤ a, b, c ≤ 100 and a b + b c + c a =
    2 2 2
    ab + bc + ca .
    Answer: 29800 This factors as ( a − b )( b − c )( c − a ) = 0. By the inclusion-exclusion principle, weget 3 · 100 − 3 · 100 + 100 = 29800.2