美国数学竟赛题

fzy

Difficulty: +++

 

N is a positive integer satisfying the following condition:

 

For any two sets of N consecutive integers A, B, we can form N pairs (a, b) with a ∈ A, b ∈ B, each number is used exactly once, such that the sums of the pairs also form a set of N consecutive integers.

 

Find all such N. 

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野 菜 花

Suppose the consecutive interger series of sums of pairs starts with m+1 then Na+(1+N)N/2 +Nb+(1+N)N/2 =N(a+b)+(m+1+m+N)N/2 Simplify it, we get N=2m-1 www.ddhw.com

fzy

You understood it corrctly. How about the other half (N = 2k)? www.ddhw.com

野 菜 花

Did I misunderstand your question? N=2k+1 (K=1,2,3,...) A={a+1, a+2,..., a+N} B={b+1,b+2,..., b+N}   Set of pairs: {(a+k,b+1), (a+k-1,b+3), (a+k-2, b+5),...,(a+1, b+N), (a+N, b+2), (a+N-1, b+4),..., (a+k+1, b+N-1) } ==> a+b+k+1, a+b+k+2, a+b+k+3,..., a+b+N+1,a+b+N+2, a+b+N+3,...,a+b+N+k N consecutive intergers   www.ddhw.com