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2017 APMO Problem 3
source link: http://siongui.github.io/2017/05/20/2017-apmo-problem-3/
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2017 APMO Problem 3
May 20, 2017
Let A(n)A(n) denote the number of sequences a1≥a2≥⋯≥aka1≥a2≥⋯≥ak of positive integers for which a1+⋯+ak=na1+⋯+ak=n and each ai+1ai+1 is a power of two. Let B(n)B(n) denote the number of sequences b1≥b2≥⋯≥bmb1≥b2≥⋯≥bm of positive integers for which b1+⋯+bm=nb1+⋯+bm=n and bj≥2bj+1bj≥2bj+1 for 1≤j≤m−11≤j≤m−1. Prove that A(n)=B(n)A(n)=B(n) for every positive integer nn.
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We'll illustrate with an example the one-to-one correspondence.
1286432168421
1286432168421
32168421
32168421
32168421
421
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1286432168421 1286432168421 32168421 32168421 32168421 421 21 21 21 21
post by Shen-Fu Tsai
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