Exercise 1803Combinatorics › Pigeonhole principleOlympiadAdded 3 Nov 2026

Difference exactly $10$

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Statement

Show that among any $11$ integers from $\{1,\ldots,100\}$ , some two differ by exactly $10$ ? — Actually false. Corrected: among any $11$ integers from $\{1,\ldots,20\}$ , some two differ by exactly $10$ .

Hint

Consider pairs $(1,11),(2,12),\ldots,(10,20)$ .

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Solution

Pairs. These $10$ pairs each differ by $10$ .

Pigeonhole. With $11$ integers and $10$ pairs, some pair contains $2$ chosen integers.

Conclusion. Two integers differ by $10$ .

Difference exactly $10$

Statement

Solution

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