HMMT 十一月 2010 · 冲刺赛 · 第 7 题

HMMT November 2010 — Guts Round — Problem 7

[ 7 ] Let f ( x, y ) = x + 2 x + y + 4 y . Let ( x , y ), ( x , y ), ( x , y ), and ( x , y ) be the vertices of a 1 1 2 2 3 3 4 4 square with side length one and sides parallel to the coordinate axes. What is the minimum value of f ( x , y ) + f ( x , y ) + f ( x , y ) + f ( x , y )? 1 1 2 2 3 3 4 4

解析

[ 7 ] Let f ( x, y ) = x + 2 x + y + 4 y . Let ( x , y ), ( x , y ), ( x , y ), and ( x , y ) be the vertices of a 1 1 2 2 3 3 4 4 square with side length one and sides parallel to the coordinate axes. What is the minimum value of f ( x , y ) + f ( x , y ) + f ( x , y ) + f ( x , y )? 1 1 2 2 3 3 4 4 Answer: -18 The square’s corners must be at ( x, y ), ( x + 1 , y ), ( x + 1 , y + 1), and ( x, y + 1) for some x and y . So, f ( x , y ) + f ( x , y ) + f ( x , y ) + f ( x , y ) 1 1 2 2 3 3 4 4 2 2 2 2 = 2( x + 2 x ) + 2(( x + 1) + 2( x + 1)) + 2( y + 4 y ) + 2(( y + 1) + 4( y + 1)) 2 2 = 4 x + 12 x + 6 + 4 y + 20 y + 10 2 2 = (2 x + 3) − 3 + (2 y + 5) − 15 ≥ − 18 3 5 This attains its minimum value of − 18 when x = − and y = − . 2 2