[/quote]
,
,
is the midpoint of 
,
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
Let 
be a positive integer. In the interior of a convex polygon there are exactly 
points of integer coordinates. Each of these points is painted red or blue. Prove that it is possible to choose 
points of the same color that are collinear.
be a positive integer. In the interior of a convex polygon there are exactly 
points of integer coordinates. Each of these points is painted red or blue. Prove that it is possible to choose 
points of the same color that are collinear.This post has been edited 3 times. Last edited by parmenides51, Sep 21, 2022, 5:17 AM
