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 2 of 5: Find when

a,b,c

all fit inside

[n,2n]

Previous step Next step

In plain words

Requiring the smallest of the triple to be at least $n$ and the largest at most $2n$ pins down exactly which $n$ work for a given $k$ .

I_k=\{n\in\mathbb{Z}: k^2+2k\le n\le 2k^2-4k\}

Detailed analysis

The triple lies in $[n,2n]$ exactly when $a\ge n$ and $c\le 2n$ , i.e. $2k^2-4k\ge n$ and $k^2+2k\le n$ . So the triple works precisely for $n$ in the interval $I_k=\{n\in\mathbb{Z}: k^2+2k\le n\le 2k^2-4k\}$ .