MathDB

(523) 6

Part of 1996 Tournament Of Towns

Problems(1)

TOT 523 1996 Autumn S A6 mathlotto - senior version

Source:

8/16/2024
The integers from 11 to 100100 are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose 1010 of these 100100 numbers. Then 10 of the integers from 11 to 100100 are drawn, and a winning ticket is one which does not contain any of them. Prove that
(a) if you buy 1313 tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket;
(b) if you buy only 1212 tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers.
(S Tokarev)
combinatorics