HMMT 二月 2013 · 冲刺赛 · 第 20 题

HMMT February 2013 — Guts Round — Problem 20

[ 11 ] The polynomial f ( x ) = x − 3 x − 4 x + 4 has three real roots r , r , and r . Let g ( x ) = 1 2 3 3 2 2 x + ax + bx + c be the polynomial which has roots s , s , and s , where s = r + r z + r z , 1 2 3 1 1 2 3 √ 2 2 − 1+ i 3 s = r z + r z + r , s = r z + r + r z , and z = . Find the real part of the sum of the 2 1 2 3 3 1 2 3 2 coefficients of g ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT 2013, 16 FEBRUARY 2013 — GUTS ROUND Organization Team Team ID# 2013

解析

[ 11 ] The polynomial f ( x ) = x − 3 x − 4 x + 4 has three real roots r , r , and r . Let g ( x ) = 1 2 3 3 2 2 x + ax + bx + c be the polynomial which has roots s , s , and s , where s = r + r z + r z , 1 2 3 1 1 2 3 √ − 1+ i 3 2 2 s = r z + r z + r , s = r z + r + r z , and z = . Find the real part of the sum of the 2 1 2 3 3 1 2 3 2 coefficients of g ( x ). 2 π 2 π 2 π i 3 2 3 Answer: − 26 Note that z = e = cos + i sin , so that z = 1 and z + z + 1 = 0. Also, 3 3 2 s = s z and s = s z . 2 1 3 1 2 Then, the sum of the coefficients of g ( x ) is g (1) = (1 − s )(1 − s )(1 − s ) = (1 − s )(1 − s z )(1 − s z ) = 1 2 3 1 1 1 2 2 3 2 3 3 3 1 − (1 + z + z ) s + ( z + z + z ) s − z s = 1 − s . 1 1 1 1 3 2 3 3 3 3 2 2 2 2 2 2 2 Meanwhile, s = ( r + r z + r z ) = r + r + r + 3 r r z + 3 r r z + 3 r r z + 3 r r z + 3 r r z + 1 2 3 2 3 3 1 1 1 1 2 3 1 1 2 2 3 2 2 3 r r z + 6 r r r . 2 1 2 3 3 1 2 3 Since the real parts of both z and z are − , and since all of r , r , and r are real, the real part of s is 1 2 3 1 2 3 9 27 3 3 3 2 2 3 r + r + r − ( r r + · · · + r r )+6 r r r = ( r + r + r ) − ( r + r + r )( r r + r r + r r )+ r r r = 2 2 1 2 3 1 2 3 1 2 3 1 2 2 3 3 1 1 2 3 1 2 3 1 3 2 2 2 9 27 3 3 − · 3 · − 4 + · − 4 = 27. 2 2 Therefore, the answer is 1 − 27 = − 26. 2013