Difference between revisions of "2021 OIM Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | Let <math>P = {p_1, p_2, \cdots p_{10}}</math> be a set of 10 different prime numbers and let <math>A</math> be the set of all integers greater than 1 such that their prime factorizations contain only primes in <math>P</math>. Each element in <math>A</math> is colored in the following way: | + | Let <math>P = \{p_1, p_2, \cdots p_{10}\}</math> be a set of 10 different prime numbers and let <math>A</math> be the set of all integers greater than 1 such that their prime factorizations contain only primes in <math>P</math>. Each element in <math>A</math> is colored in the following way: |
a) each element in <math>P</math> has a distinct color, | a) each element in <math>P</math> has a distinct color, | ||
| − | b) if <math>m, n \in A</math>, then <math>mn</math> has the same color as <math>m</math> or <math>n</math>, | + | b) if <math>m, n \in A</math>, then <math>mn</math> has the same color as either <math>m</math> or <math>n</math>, |
| − | c) for each pair of distinct colors <math>R</math> and <math>S</math>, there are no <math>j, k, m, n \in A</math> (not necessarily distinct), with <math>j, k</math> colored <math>R</math> and <math>m</math>, <math>n</math> colored <math>S</math>, such that both <math>j</math> divides <math>m</math> and <math>n</math> divides <math>k</math>. | + | c) for each pair of distinct colors <math>R</math> and <math>S</math>, there are no <math>j, k, m, n \in A</math> (not necessarily distinct), with <math>j, k</math> colored <math>R</math> and <math>m</math>, <math>n</math> colored <math>S</math>, such that both <math>j</math> divides <math>m</math> and <math>n</math> divides <math>k</math>. |
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| + | Show that there is some prime in <math>P</math> such that all of its multiples in <math>A</math> have the same color. | ||
== Solution == | == Solution == | ||
Latest revision as of 16:07, 23 March 2025
Problem
Let 
be a set of 10 different prime numbers and let 
be the set of all integers greater than 1 such that their prime factorizations contain only primes in 
. Each element in is colored in the following way:
a) each element in has a distinct color,
b) if 
, then 
has the same color as either 
or 
,
c) for each pair of distinct colors 
and 
, there are no 
(not necessarily distinct), with 
colored and , colored , such that both 
divides and divides 
.
Show that there is some prime in such that all of its multiples in have the same color.
Solution
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