IMO 2021 Q1 #
n≥100 be an integer. Ivan writes the numbers
n, n+1,..., 2n each on different cards.
He then shuffles these
n+1 cards, and divides them into two piles. Prove that at least one
of the piles contains two cards such that the sum of their numbers is a perfect square.
We show there exists a triplet
a, b, c ∈ [n , 2n] with
a < b < c and each of the sums
(a + b),
(b + c),
(a + c) being a perfect square. Specifically, we consider the linear system of
a + b = (2 * l - 1) ^ 2 a + c = (2 * l) ^ 2 b + c = (2 * l + 1) ^ 2
which can be solved to give
a = 2 * l * l - 4 * l b = 2 * l * l + 1 c = 2 * l * l + 4 * l
Therefore, it is enough to show that there exists a natural number l such that
n ≤ 2 * l * l - 4 * l and
2 * l * l + 4 * l ≤ 2 * n for
n ≥ 100.
Then, by the Pigeonhole principle, at least two numbers in the triplet must lie in the same pile, which finishes the proof.