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1+2+... +a_1\le 1/3(1+2+... +n) and 1+2+...+(a_1+a_2) \le 2/3(1+2+... +n)

Source: Rioplatense Olympiad 2014 level 3 P6

September 6, 2018

Setspartitionnumber theoryinequalities

Problem Statement

Let

n \in N

such that

1 + 2 + ... + n

is divisible by

3

. Integers

a_1\ge a_2\ge a_3\ge 2

have sum

n

and they satisfy

1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n )

and

1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )

. Prove that there is a partition of

\{ 1 , 2 , ... , n\}

in three subsets

A_1, A_2, A_3

with cardinals

| A_i| = a_i, i = 1 , 2 , 3

, and with equal sums of their elements .