HMMT 十一月 2010 · 团队赛 · 第 8 题

HMMT November 2010 — Team Round — Problem 8

[ 4 ] A function f ( x, y ) is linear in x and in y . f ( x, y ) = for x, y ∈ { 3 , 4 } . What is f (5 , 5)? xy 1

解析

[ 4 ] A function f ( x, y ) is linear in x and in y . f ( x, y ) = for x, y ∈ { 3 , 4 } . What is f (5 , 5)? xy 1 Answer: The main fact that we will use in solving this problem is that f ( x + 2 , y ) − f ( x + 1 , y ) = 36 f ( x + 1 , y ) − f ( x, y ) whenever f is linear in x and y . Suppose that f ( x, y ) = axy + by + cx + d = x ( ay + c ) + ( by + d ) for some constants a , b , c , and d . Then it is easy to see that f ( x + 2 , y ) − f ( x + 1 , y ) = ( x + 2)( ay + c ) + ( by + d ) − ( x + 1)( ay + c ) − ( by + d ) = ay + c f ( x + 1 , y ) − f ( x, y ) = ( x + 1)( ay + c ) + ( by + d ) − x ( ay + c ) − ( by + d ) = ay + c, which implies that f ( x + 2 , y ) − f ( x + 1 , y ) = f ( x + 1 , y ) − f ( x, y ). In particular, f (5 , y ) − f (4 , y ) = f (4 , y ) − f (3 , y ), so f (5 , y ) = 2 f (4 , y ) − f (3 , y ). Similarly, f ( x, 5) = 2 f ( x, 4) − f ( x, 3). Now we see that: f (5 , 5) = 2 f (5 , 4) − f (5 , 3) = 2[2 f (4 , 4) − f (3 , 4)] − [2 f (4 , 3) − f (3 , 3)] = 4 f (4 , 4) − 2 f (3 , 4) − 2 f (4 , 3) + f (3 , 3) 4 4 1 = − + 16 12 9 1 1 1 = − + 4 3 9 1 1 = − 9 12 1 = , 36 1 so the answer is . 36 1