2016 UMO Problems/Problem 3

Problem

Can each positive integer $1,2,3,\ldots$ be colored either red or blue, such that for all positive integers $a,b,c,d$ (not necessarily distinct), if $a + b + c = d$ then $a,b,c,d$ are not all the same color?

Solution

Solution We claim that no such coloring exists. For the sake of contradiction, assume that such a coloring exists. Without loss of generality, we may assume that 1 is colored red. In the string of steps below, if a number n is known to be colored red, we denote it as $n^R$ , and if it is known to be colored blue, we denote it as $n^B$ . Each of the implications below follow from the fact that if all but one of the distinct numbers in $a + b + c = d$ are the same color, then the ﬁnal number must be a different color. $1^R + 1^R + 1^R = 3 \Rightarrow 3$ is blue. $3^B + 3^B + 3^B = 9 \Rightarrow 9$ is red. $1^R + 4 + 4 = 9^R \Rightarrow 4$ is blue. Now we have $1^R + 1^R + 9^R = 11 \Rightarrow 11$ is blue AND $3^B + 4^B + 4^B = 11 \Rightarrow 11$ is red. This is a contradiction because $11$ can only have one color. Therefore, no such coloring exists.

2016 UMO (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6
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2016 UMO Problems/Problem 3

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