be a cyclic quadrilateral. A circle centered at 
passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
)
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle 
and 
are similar.
such that 
, where 
are non-negative reals such that 
.
.
be the second intersection of the circumcircle of triangle 
with the internal bisector of the angle 
.
be the circumcenter of triangle 
and 
the orthocenter of triangle 
.
is equal to the circumradius of triangle 
such that the following property holds: if we list the divisors of 
in increasing order as 
,![\[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \]](http://latex.artofproblemsolving.com/a/d/c/adc94af88b3579164aabba51bd021442133e364a.png)
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
be the incircle of a fixed equilateral triangle 
be a variable line that is tangent to 
at points 
,
is chosen such that 
and 
.
be positive real numbers such that ![\(a+b+c=4\sqrt[3]{abc}\)](http://latex.artofproblemsolving.com/2/8/2/28214a1814762b71858db1543fae1a50212eaafa.png)
.![\[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]](http://latex.artofproblemsolving.com/7/0/4/7043a170ac5535465a5a1b4daccb1f1de3043280.png)
are multiples of 
or 
but not 
?
![\[222{,}222-22{,}222-2{,}222-222-22-2?\]](http://latex.artofproblemsolving.com/1/5/6/156c3662fe4c874a80e54887f76c302c13b9d14c.png)
Y by Adventure10
Determine all positive integers 
satisfying the condition that there exists a unique positive integer 
such that there exists a rectangle which can be decomposed into congruent squares and can also be decomposed into 
congruent squares.
satisfying the condition that there exists a unique positive integer 
such that there exists a rectangle which can be decomposed into congruent squares and can also be decomposed into 
congruent squares.This post has been edited 1 time. Last edited by ayan.nmath, Jul 17, 2019, 1:03 PM
,
,
,
of 
,
.
or more odd divisors, then it can not satisfy the condition:every positive integer 
and 
and 
are clearly coprime. So we only need to consider numbers 
,
,
.
