HMMT 十一月 2010 · 团队赛 · 第 9 题

HMMT November 2010 — Team Round — Problem 9

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

解析

[ 5 ] A function f ( x, y, z ) is linear in x , y , and z such that f ( x, y, z ) = for x, y, z ∈ { 3 , 4 } . What is xyz f (5 , 5 , 5)? 1 Answer: We use a similar method to the previous problem. Notice that f ( x, y, 5) = 2 f ( x, y, 4) − 216 f ( x, y, 3). Let f denote the function from the previous problem and f the function from this problem. 2 3 Team Round 1 Since 3 f ( x, y, 3) is linear in x and y , and 3 f ( x, y, 3) = for all x, y ∈ { 3 , 4 } , the previous problem 3 3 xy 1 implies that 3 f (5 , 5 , 3) = = f (5 , 5). Similarly, 4 f (5 , 5 , 4) = f (5 , 5). Now we have 3 2 3 2 36 f (5 , 5 , 5) = 2 f (5 , 5 , 4) − f (5 , 5 , 3) 3 3 3 1 1 = f (5 , 5) − f (5 , 5) 2 2 2 3 1 = f (5 , 5) 2 6 1 = 6 · 36 1 = . 216 1