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.
Problem 892
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 “ba is greater than all numbers to the right, in particular, greater than b$ ).