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Let

A(n)

denote the number of sequences

a_1\ge a_2\ge\cdots{}\ge a_k

of positive integers for which

a_1+\cdots{}+a_k = n

and each

a_i +1

is a power of two

(i = 1,2,\cdots{},k)

. Let

B(n)

denote the number of sequences

b_1\ge b_2\ge \cdots{}\ge b_m

of positive integers for which

b_1+\cdots{}+b_m =n

and each inequality

b_j\ge 2b_{j+1}

holds

(j=1,2,\cdots{}, m-1)

. Prove that

A(n) = B(n)

for every positive integer

n

.
Senior Problems Committee of the Australian Mathematical Olympiad Committee

Problem Statement