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by Hip1zzzil, Mar 29, 2025, 10:23 AM
An acute triangle 
is given.
is the interior and the incircle of meets 
at 
. 
and 
meet at 
. Let 
be a tangent from D to the circumcircle of 
, and define 
and 
on 
and 
, respectively.
Prove
meet at one point.
is given.
is the interior and the incircle of meets 
at 
. 
and 
meet at 
. Let 
be a tangent from D to the circumcircle of 
, and define 
and 
on 
and 
, respectively.Prove
meet at one point.This post has been edited 3 times. Last edited by Hip1zzzil, 41 minutes ago
Reason: Typo
Reason: Typo
that satisfies:
or 
be positive real numbers such that 
.
satisfies the following condition.
,![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
,
.
satisfies the equation:![$\sum_{k=1}^{n}\frac{1}{2}(1-(-1)^{[\frac{n}{k}]})a_{k}=1$](http://latex.artofproblemsolving.com/3/2/1/3216129b3e42dfb49eb8bb764fafa1b9fbf28882.png)
,
.
is the foot of the altitude from 
of triangle 
.
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles 
.
.
and 
are divisible by 
.
,
is the largest prime divisor of 
be circumcenter of a non-isosceles triangle 
.
be foots of perpendicular lines from 
.
is cyclic and 
is the circumcenter of 
.
bisects 
denote the set of all positive rational numbers. Determine all functions 
satisfying 
for all 
March 2025
February 2025