AoPS Academy
with incenter 
. The incircle of triangle is tangent to 
at 
. Let 
and 
be points on the side BC such that 
and 
, respectively. Let 
and 
be the incenter of triangles 
and 
, respectively. Prove that 
is the Euler line of triangle 
.
, 
, 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
and 
be positive real numbers with 
. Prove that
For which and equality holds?
, find the number of terms with coefficient divisible by 
.
and of an equilateral triangle are taken points and , and on the side 
, points 
and 
so that 
. Prove that the angle between lines 
and 
is equal to 
.
over 
. Ignore "pathological" solutions.[/quote]
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
of the pizza, where 
is a positive integer. Find .
be an isosceles triangle with 
. Let be the foot of the alttiude from 
, and 
be the external angle bisector at . Take a point 
on so that 
, on the same side as 
wrt 
. The bisector of the angle 
cuts 'at 
. Show that 
be an isosceles triangle with . Let be the foot of the alttiude from , and be the external angle bisector at . Take a point on so that , on the same side as wrt . The bisector of the angle cuts 'at . Show that
such that 
for all 
