HMMT 二月 2007 · TEAM2 赛 · 第 7 题
HMMT February 2007 — TEAM2 Round — Problem 7
题目详情
英文原题
- [ 20 ] Three positive reals x, y, and z are such thatx + 2( y − 1)( z − 1) = 852
y + 2( z − 1)( x − 1) = 842
z + 2( x − 1)( y − 1) = 89 .2
Compute x + y + z.
( ) ( ) ( )
2 2 21
解析
英文解析
- [ 20 ] Three positive reals x, y, and z are such thatx + 2( y − 1)( z − 1) = 852
y + 2( z − 1)( x − 1) = 842
z + 2( x − 1)( y − 1) = 89 .2
Compute x + y + z.
Answer: 18 . Add the three equations to obtain
2 2 2
x + y + z + 2 xy + 2 yz + 2 zx − 4 x − 4 y − 4 z + 6 = 258 ,
which rewrites as ( x + y + z − 2) = 256 . Evidently, x + y + z = 2 ± 16 . Since x, y, and z are positive,2
x + y + z > 0 so x + y + z = 2 + 16 = 18 .
( ) ( ) ( )
2 2 21