HMMT 十一月 2010 · 冲刺赛 · 第 7 题
HMMT November 2010 — Guts Round — Problem 7
题目详情
英文原题
- [ 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 off ( x , y ) + f ( x , y ) + f ( x , y ) + f ( x , y )?
1 1 2 2 3 3 4 4
解析
英文解析
- [ 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 off ( 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 somex 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