Problem 1

Let

n \ge 100

be an integer. Ivan writes the numbers

n, n+1, \ldots, 2n

each on a different card. He then shuffles these

n+1

cards and divides them into two piles. Prove that at least one of the piles contains two cards whose numbers sum to a perfect square.

Step 1 of 5: Construct a triple with three square pairwise sums

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In plain words

Three consecutive squares $(2k-1)^2,(2k)^2,(2k+1)^2$ can be realized simultaneously as the three pairwise sums of a triple of integers.

a=2k^2-4k,\quad b=2k^2+1,\quad c=2k^2+4k

Detailed analysis

Set $a+b=(2k-1)^2$ , $b+c=(2k+1)^2$ , $c+a=(2k)^2$ . Solving this linear system gives $a=2k^2-4k$ , $b=2k^2+1$ , $c=2k^2+4k$ . For the range used below, $k\ge9$ , these satisfy $a<b<c$ .