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HMMT 二月 2008 · 冲刺赛 · 第 33 题

HMMT February 2008 — Guts Round — Problem 33

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 18 ] Let a , b , c be nonzero real numbers such that a + b + c = 0 and a + b + c = a + b + c . Find 2 2 2 the value of a + b + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND
解析
  1. [ 18 ] Let a , b , c be nonzero real numbers such that a + b + c = 0 and a + b + c = a + b + c . Find 2 2 2 the value of a + b + c . 6 Answer: Let σ = a + b + c , σ = ab + bc + ca and σ = abc be the three elementary symmetric 1 2 3 5 3 3 3 polynomials. Since a + b + c is a symmetric polynomial, it can be written as a polynomial in σ , σ 1 2 and σ . Now, observe that σ = 0, and so we only need to worry about the terms not containing σ . By 3 1 1 3 3 3 considering the degrees of the terms, we see that the only possibility is σ . That is, a + b + c = kσ 3 3 for some constant k . By setting a = b = 1, c = − 2, we see that k = 3. 5 5 5 By similar reasoning, we find that a + b + c = hσ σ for some constant h . By setting a = b = 1 and 2 3 c = − 2, we get h = − 5. 9 So, we now know that a + b + c = 0 implies 3 3 3 5 5 5 a + b + c = 3 abc and a + b + c = − 5 abc ( ab + bc + ca ) 3 3 3 5 5 5 Then a + b + c = a + b + c implies that 3 abc = − 5 abc ( ab + bc + ca ). Given that a, b, c are nonzero, 3 6 2 2 2 2 we get ab + bc + ca = − . Then, a + b + c = ( a + b + c ) − 2( ab + bc + ca ) = . 5 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND