In triangle ABC prove that:
by khanhsy, Mar 29, 2025, 11:12 AM
Functional Equation!
by EthanWYX2009, Mar 29, 2025, 10:48 AM
Find all functions 
such that 
is unbounded and
![\[2f(m)f(n)-f(n-m)-1\]](//latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
Another "OR" FE problem
by pokmui9909, Mar 29, 2025, 10:16 AM
Let 
be the set of real numbers. Find all functions 
that satisfy the following condition. Here, 
is the function obtained by composing 
times, that is, 
(Condition) For all
, 
be the set of real numbers. Find all functions 
that satisfy the following condition. Here, 
is the function obtained by composing 
times, that is, 
(Condition) For all
, 
This post has been edited 1 time. Last edited by pokmui9909, an hour ago
Functions OTL
by Hip1zzzil, Mar 29, 2025, 10:11 AM
Find all functions 
that satisfies:
or 
that satisfies:
or 
This post has been edited 1 time. Last edited by Hip1zzzil, 2 hours ago
Reason: Typo
Reason: Typo
They copied their problem!
by pokmui9909, Mar 29, 2025, 10:03 AM
Sequence 
satisfies the following condition.
(Condition) For all positive integer
, ![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](//latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
For a positive integer
, compute 
.
satisfies the following condition.(Condition) For all positive integer
, ![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.For a positive integer
, compute 
.This post has been edited 5 times. Last edited by pokmui9909, an hour ago
Let $a,b,c$ be nonnegative real numbers. Prove that
by khanhsy, Mar 29, 2025, 9:21 AM
Let 
be nonnegative real numbers such that 
. Prove that:
be nonnegative real numbers such that 
. Prove that:
This post has been edited 3 times. Last edited by khanhsy, 3 hours ago
Reason: edit
Reason: edit
Proving ZA=ZB
by nAalniaOMliO, Mar 28, 2025, 8:36 PM
Point 
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
Prove that
.
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.Prove that
.Two numbers divisible by 2025
by nAalniaOMliO, Mar 28, 2025, 8:35 PM
Find the smallest positive integer such that both 
and 
are divisible by 
.
such that both 
and 
are divisible by 
.This post has been edited 1 time. Last edited by nAalniaOMliO, an hour ago
Not so classic orthocenter problem
by m4thbl3nd3r, Mar 28, 2025, 4:59 PM
Let 
be circumcenter of a non-isosceles triangle and be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and 
is the circumcenter of , 
. Prove that 
bisects 
be circumcenter of a non-isosceles triangle and be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and 
is the circumcenter of , 
. Prove that 
bisects 
Attachments:
This post has been edited 2 times. Last edited by m4thbl3nd3r, Yesterday at 5:00 PM
Rational FEs
by Functional, Jul 17, 2019, 12:12 PM
Let 
denote the set of all positive rational numbers. Determine all functions 
satisfying 
for all 
denote the set of all positive rational numbers. Determine all functions 
satisfying 
for all 
This post has been edited 1 time. Last edited by Functional, Jul 17, 2019, 12:12 PM
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