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up vote 12 down vote favorite 2 A book consists of 100 pages and contains 100 lemmas and some images. Each lemma is at most one page long and can't be split into two pages (it has to fit in one page). The lemmas are numbered from 1 to 100 and are written in ascending order. Prove that there must be at least one lemma written on a page with the same number as the lemma's number. If lemma no 1 is written on page no 1, then it is proved. Let's assume lemma nr 1 is written on page nr k, k>1. Then in at least one page there must be 2 lemmas. Let's assume that always in page k+i we have the lemma nr i+1 and so on. Then the last 100-k-i lemmas must fit in the last page, which means that there will be at least one lemma (number 100) in page 100. But I don't know how to express it in a more mathematical way! A...