[/quote]
,
and 
Find minimum 
is inscribed into a circle 
.
be the centroid of 
,
be an altitude of this triangle. A ray 
meets 
.
is tangent to 
.
denote the set of positive integers. For any positive integer 
,
is called 
for all 
.
be real numbers such that 
Prove that
Let 
Prove that
Prove that
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
and 
.
and 
.
and 
intersect at point 
be the reflection of 
and 
intersect at point 
,
.
.
such that for all positive integer 
,
Where 
denotes the number of divisors of 
.
,
such that for all positive integer 
.
then![\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]](http://latex.artofproblemsolving.com/6/d/1/6d1cd28a47704aa0cac2a6c6b572816fc1db790a.png)
Generalize the problem.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
be the intersections of 
with the sides of 
are the midpoints of 
,
,
and 
.
![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
Y by nguyendangkhoa17112003, Adventure10, Mango247
Let there be an acute triangle 
with incenter 
. 
is the foot of the perpendicular from to 
. The line which passes through 
and is perpendicular to 
hits line 
at 
. The line which passes through and is perpendicular to hits the line which passes through 
and is perpendicular to at 
. Prove that 
are colinear.
with incenter 
. 
is the foot of the perpendicular from to 
. The line which passes through 
and is perpendicular to 
hits line 
at 
. The line which passes through and is perpendicular to hits the line which passes through 
and is perpendicular to at 
. Prove that 
are colinear.
. Quite a few cevians.
.
and 
.![\[\frac{AK}{CL} = \frac{\tan \frac{C}{2}}{\tan \frac{A}{2}} = \frac{\frac{IE}{EC}}{\frac{IE}{EA}} = \frac{EA}{EC}\qquad\square\]](http://latex.artofproblemsolving.com/1/2/b/12bb33aca03d3444977bde540630fa33913c42e9.png)
be the reflections of 
be the reflection of 
and 
.
and 
.
we finish by showing that ![\[\left| \begin{matrix}a+b-c & 0 &-a+b+c \\ a(a-c) & b(a-c) & bc \\ ab & b(c-a) &(a-b)(c-a) \end{matrix} \right|=0.\]](http://latex.artofproblemsolving.com/e/4/f/e4fc43cb800ce72dbdf6b02d5c5d68840aa29a8a.png)
and a lot of easy angles - so simply just length chase. This is done above.
touch 
at 
.
at 
,
are conllinear.
are conllinear.
.
