MathLabs

Problem 1

Let n≥100n \ge 100 be an integer. Ivan writes the numbers n,n+1,…,2nn, n+1, \ldots, 2n each on a different card. He then shuffles these n+1n+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 4 of 5: Show consecutive intervals overlap for k≥9k\ge9
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In plain words

If the right end of IkI_k reaches at least one less than the left end of Ik+1I_{k+1}, the intervals leave no integer gap between them.

2k2−4k≥(k+1)2+2(k+1)−12k^2-4k\ge (k+1)^2+2(k+1)-1
Detailed analysis

For k≥9k\ge9, one checks 2k2−4k≥(k+1)2+2(k+1)−12k^2-4k\ge (k+1)^2+2(k+1)-1 (equivalently k2−8k−2≥0k^2-8k-2\ge0, true once k≥9k\ge9), so the right end of IkI_k reaches at least the integer just before the left end of Ik+1I_{k+1}; hence I9,I10,I11,…I_9,I_{10},I_{11},\ldots cover every integer from 9999 onward with no gaps.