2024 USAMO Problems/Problem 1
Find all integers such that the following property holds: if we list the divisors of
in increasing order as
, then we have
Solution (Explanation of Video)
We can start by verifying that and
work by listing out the factors of
and
. We can also see that
does not work because the terms
, and
are consecutive factors of
. Also,
does not work because the terms
, and
appear consecutively in the factors of
.
Note that if we have a prime number and an integer
such that both
and
are factors of
, then the condition cannot be satisfied.
If is odd, then
is a factor of
. Also,
is a factor of
. Since
for all
, we can use Bertrand's Postulate to show that there is at least one prime number
such that
. Since we have two consecutive factors of
and a prime number between the smaller of these factors and
, the condition will not be satisfied for all odd $n\geq7%.
If$ (Error compiling LaTeX. Unknown error_msg)n\geq8(2)(\frac{n-2}{2})(n-2)=n^2-4n+4
n!
(n-3)(n-1)=n^2-4n+3
n!
2n<n^2-4n+3
n\geq8
p
n<p<n^2-4n+3
n!
n
n\geq8$.
Therefore, the only numbers that work are$ (Error compiling LaTeX. Unknown error_msg)n=3n=4$.
~alexanderruan
Video Solution
https://youtu.be/ZcdBpaLC5p0 [video contains problem 1 and problem 4]