MathDB
encircling 999 numbers with red, green, blue chalk

Source: Dutch NMO 2016 p1

September 7, 2019
combinatoricsColoring

Problem Statement

(a) On a long pavement, a sequence of 999999 integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles 500500 of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500. Next, Jeroen encircles 500500 of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers 500,499,498,...,2,1500, 499, 498, ...,2,1.
Prove that the middle number in the sequence of 999999 numbers is circled both in red and in blue.
(b) Merlijn and Jeroen cross the street and find another sequence of 999999 integers on the pavement. Again Merlijn circles 500500 of the numbers with red chalk. Again the numbers circled in red are precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500 from left to right. Now Jeroen circles 500500 of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500 from left to right.
Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of 999999 numbers.
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