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Number Theory Level 2

We colored each of the numbers from $1, 2, 3, \dots, 32, 33$ We used $11$ colors, and each color was used for exactly $3$ numbers. Is it possible that for the numbers with the same color ( $a<b<c$ ) $a+b=c$ is always true?

No, it's not possible. Yes, it's possible.

1 solution

Áron Bán-Szabó
Jul 9, 2017

Suppose it is possible. Then if $x, y$ and $z$ are colored the same, then: $x+y=z$

$\Rightarrow x+y+z=2z\equiv 0 \ \mod2$

So the sum of numbers with same color is always divisible by $2$ , and the sum of the numbers from $1$ to $33$ has to be even, but this leads to controversy, because: $1+2+3+\dots+33=\dfrac{33*34}{2}=33*17\equiv1 \ \mod2$

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