HMMT 十一月 2010 · 团队赛 · 第 9 题

英文解析

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 isxyzf (5 , 5 , 5)?
   Answer: We use a similar method to the previous problem. Notice that f ( x, y, 5) = 2 f ( x, y, 4) −1
   f ( x, y, 3). Let f denote the function from the previous problem and f the function from this problem.216
   2 3
   Team Round
   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 1
   3 3
   1 xyimplies 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
   f (5 , 5 , 5) = 2 f (5 , 5 , 4) − f (5 , 5 , 3)36
   3 3 3
   1 1 = f (5 , 5) − f (5 , 5)
   2 2
   2 3 = f (5 , 5)1 =162
   6 · 36 = .1
   1216