Track / Stage 5 / 277 of 400 #877 of 1964

Problem 877

AIME late

Combinatorics Difficulty 5.6 Prove it

4. A row of 101 cards is laid out. On each of the 50 cards lying in this row at even positions, a symbol > or < is drawn. Prove that, no matter how these symbols are drawn, the remaining cards can be filled with the numbers 1, 2, ... 51 (using each exactly once) so that all the resulting inequalities are true.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

First solution. First, write a number above all the cards lying in odd positions. Above the first card, write the number 0. Then we will move to the right and each time after the sign “>” write a number that is 1 less than the previous one, and after the sign “b $the inequality is true (since by construction$ a $is greater than all numbers to the right, in particular, greater than$ b$ ).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.