2024 IMO Problems/Problem 2
Find all positive integer pairs such that there exists positive integer
holds for all integer
.
Contents
[hide]Solution 1
We will determine all pairs of positive integers such that
for all
.
First, works with
. Now for any solution
:
\begin{lemma}
or
.
\end{lemma}
\begin{proof}
Since divides both
and
, it divides their difference
. Similarly,
divides
. Thus
divides
, so
divides
. Hence
divides
, a contradiction unless
divides both
and
.
\end{proof}
Let and write
,
with
. Then
Using Euler's theorem, for where
, we have:
Similarly, . Since these are divisible by
, and
must divide
, we must have
, giving
.
~brandonyee
Video Solution
https://www.youtube.com/watch?v=VXFG1t_ksfI (including motivation to derive solution)
Video Solution(Fermat's little theorem,In English)
Video Solution(Fermat's little theorem,In Chinese)
Video Solution
See Also
2024 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |