Trigonometric Product
by Henryfamz, May 13, 2025, 4:52 PM
Compute 
Gcd(m,n) and Lcm(m,n)&F.E.
by Jackson0423, May 13, 2025, 4:12 PM
Find all functions 
such that for all positive integers 
,
![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](//latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
, respectively.
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
, respectively.3 variable FE with divisibility condition
by pithon_with_an_i, May 13, 2025, 4:11 PM
Find all functions 
such that 
for all 
.
such that 
for all 
.This post has been edited 1 time. Last edited by pithon_with_an_i, an hour ago
Reason: Typo
Reason: Typo
f(f(n))=2n+2
by Jackson0423, May 13, 2025, 4:07 PM
Let be a function satisfying the following conditions for all 
:
![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](//latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
be a function satisfying the following conditions for all 
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.Tangents involving a centroid with an isosceles triangle result
by pithon_with_an_i, May 13, 2025, 4:06 PM
A triangle 
has centroid 
. A line parallel to 
passing through intersects the circumcircle of at a point 
. Let lines 
and intersect at 
. Suppose a point 
is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at 
and concur. Prove that 
.
has centroid 
. A line parallel to 
passing through intersects the circumcircle of at a point 
. Let lines 
and intersect at 
. Suppose a point 
is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at 
and concur. Prove that 
.
Thailand MO 2025 P2
by Kaimiaku, May 13, 2025, 7:38 AM
A school sent students to compete in an academic olympiad in 
differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.Thailand MO 2025 P3
by Kaimiaku, May 13, 2025, 6:48 AM
Let 
be positive real numbers such that 
. Prove that 
be positive real numbers such that 
. Prove that 
Gives typical russian combinatorics vibes
by Sadigly, May 8, 2025, 4:15 PM
You are given a positive integer 
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
Prove that, no matter what, the person standing on the coordinates will not have more than 
litres of water.
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.Prove that, no matter what, the person standing on the coordinates
will not have more than 
litres of water.This post has been edited 2 times. Last edited by Sadigly, May 11, 2025, 6:40 AM
Beautiful numbers in base b
by v_Enhance, Oct 21, 2023, 11:00 PM
A positive integer is called beautiful if, for every integer 
, the base-
representation of contains the consecutive digits , 
, , 
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
Oleg Kryzhanovsky
is called beautiful if, for every integer 
, the base-
representation of contains the consecutive digits , 
, , 
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.Oleg Kryzhanovsky
This post has been edited 1 time. Last edited by v_Enhance, Oct 22, 2023, 11:43 PM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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