91年全國高中數學能力競賽試題

Question

設對所有的非負整數x,y,z，函數 f 滿足下列兩個條件：
(1)f(x,y)= - f(y,x)
(2)f(x+y,z)+f(y+z,x)+f(z+x,y)=0
試證：f(x,y)=0

Answer 1

let x=y=z=0 so f(0,0)=0
x=y=0 so f(z,0)=f(0,z)=0
let x=k , y=N-1-k , z= 1 0<=k<=N-1
so T=f (N-1,1)+ f (N-k,k) +f (k+1,N-1-k)=0
(sigma k=0 to N-1) T=0
that(sigma k=0 to N-1)f (k+1,N-1-k)=- (sigma k=0 to N-1) f (N-k,k)-f(0,N)
so f(N-k,k)=f(N-k-1,k+1)
so f(N-k,k)=0 k=0...N

2006-03-04 16:50:41 補充：
sorry line 6that(sigma k=0 to N-1)f (k+1,N-1-k)=- (sigma k=0 to N-1) f (N-k,k)-f(0,N)=>that(sigma k=0 to N-1)f (k+1,N-1-k)=- (sigma k=1 to N-1) f (N-k,k)-f(0,N)

Answer 2

ＡＭＣ？