2021 OIM Problems/Problem 1

Problem

Let $P = {p_1, p_2, \cdots p_{10}}$ be a set of 10 different prime numbers and let $A$ be the set of all integers greater than 1 such that their prime factorizations contain only primes in $P$ . Each element in $A$ is colored in the following way:

a) each element in $P$ has a distinct color,

b) if $m, n \in A$ , then $mn$ has the same color as $m$ or $n$ ,

c) for each pair of distinct colors $R$ and $S$ , there are no $j, k, m, n \in A$ (not necessarily distinct), with $j, k$ colored $R$ and $m$ , $n$ colored $S$ , such that both $j$ divides $m$ and $n$ divides $k$ . Show that there is some prime in $P$ such that all of its multiples in $A$ have the same color.

Solution

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2021 OIM Problems/Problem 1

Problem

Solution

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