MathDB

(a) On a long pavement, a sequence of

999

integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles

500

of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers

1, 2, 3, ...,499, 500

. Next, Jeroen encircles

500

of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers

500, 499, 498, ...,2,1

.
Prove that the middle number in the sequence of

999

numbers is circled both in red and in blue.
(b) Merlijn and Jeroen cross the street and find another sequence of

999

integers on the pavement. Again Merlijn circles

500

of the numbers with red chalk. Again the numbers circled in red are precisely the numbers

1, 2, 3, ...,499, 500

from left to right. Now Jeroen circles

500

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, 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

999

numbers.

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