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a^2_1+a^_22+...+a^2_k is divisible by a_1+a_2+...+a_k

Source: All-Russian olympiad 1995, Grade 11, Second Day, Problem 5

October 21, 2013

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Problem Statement

Prove that for every natural number

a_1>1

there exists an increasing sequence of natural numbers

a_n

such that

a^2_1+a^2_2+\cdots+a^2_k

is divisible by

a_1+a_2+\cdots+a_k

for all

k \geq 1

.
A. Golovanov