2025 Caucasus MO Juniors P3

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2025 Caucasus MO Juniors P3

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Source: Caucasus MO

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#1 h Mar 26, 2025, 12:56 AM

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Let $K$ be a positive integer. Egor has $100$ cards with the number “ $2$ ” written on them, and $100$ cards with the number “ $3$ ” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to $K$ . Find the greatest $K$ such that Egor will not be able to paint the cards in such a way.

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FarrukhKhayitboyev

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FarrukhKhayitboyev

#2 h Apr 4, 2025, 8:49 AM

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Answer : $246$ .
Claim $1$ : anything above $250$ is "paintable".
Proof: Paint $50$ cards of $2$ in one color and other in the opposite color, do the same with $3$ 's
The biggest mono-colored set is $250$ , so anything above $250$ is "paintable".
Claim $2$ : $246$ is always "unpaintable".
Proof: Let's take a color whose sum is over $250$ , if the sum is $0$ mod $3$ , then we will remove $3$ 's till we get $246$ and
it is possible because if we ran out of $3$ 's then the sum would be maximum $200$ , contradiction, we won't ran out of $3$ 's.
If the sum is $1$ mod $3$ then we will eliminate two $2$ 's and proceed like the first case.
If the sum is $2$ mod $3$ . Then we will eliminate one $2$ and proceed like the first case.
Now we should prove that $247,248,249,250$ are "paintable".
Obviously, there is case for $249$ (left as an exercise for the reader.).
Let's look at the case where all $2$ 's are red and all $3$ 's are blue, then the sum should divide $3$ , and $247,248,250$ are not divisible by $3$ .
And we are done.

This post has been edited 1 time. Last edited by FarrukhKhayitboyev, Apr 6, 2025, 3:21 AM

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