AoPS Academy
that satisfy the equation
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)
'is a perfect square.
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)
. To conclude, observe that![\[0=\det\begin{pmatrix}1&0&0\\b-c&c-a&a-b\\1&1&1\end{pmatrix}\iff 2c=a+b.\]](http://latex.artofproblemsolving.com/4/0/e/40e4e2916bb39e5367e94573864b3fe0b7d687e7.png)
as the base of the triangle. 
is its vertical height, 
its inradius, and 
the vertical distance from the base to its centroid.
be twice the area of the triangle. Then 
and also 
we have 
, so 
such that 
for all 
of a non-equilateral triangle satisfy 
,