### APMO 2017 P3

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**Sat May 27, 2017 1:36 pm**Let $A(n)$ denote the number of sequences $a_1 \geq a_2 \geq ....\geq a_k$ of positive integers for which $a_1 + a_2 + ... + a_k = n$ and each $a_i + 1$ is a power of two . let $B(m)$ denote the number of sequences $b_1 \geq b_2 \geq ....\geq b_m$ of positive integers for which $b_1 + b_2 + .... b_m = n$ and each inequality $b_j \geq 2b_{j+1}$ holds $(j = 1,2,...,m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$.