2024 SSMO Accuracy Round Problems/Problem 3
Problem
Three distinct random integers ,
, and
are selected so that
. Let the probability that
be
, where
. Find
.
Solution
Note that have residues
when taken modulo 5. We have 2 cases for residues modulo 5: either one is 0, one is 1, and one is 4, or they are all 0. For the first case, we have
ways to choose a
, 1 way to choose a
ways to choose a
and
ways to permute them, giving
solutions. For the other case, we have
ways to choose the 3 multiples of 5. Thus, the total number of solutions is
meaning the answer is
~SMO_Team