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 3 of 5: Check the interval starts covering

n=99

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I_9=\{99,100,\ldots,126\}

Detailed analysis

For $k=9$ , $I_9=\{99,100,\ldots,126\}$ , so every $n$ with $99\le n\le126$ — in particular every $n\ge100$ in this range — already admits a valid triple.