[/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.
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
.
(not containing 
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 Adventure10, Mango247
Let 
be the incenter of triangle 
, and let 
, 
, and 
be arbitrary points lying on segments 
,
, and 
, respectively. The perpendicular bisectors of segments 
, 
, and 
form triangles 
. Prove that the circumcenter of triangle coincides with the circumcenter of triangle if and only if is the orthocenter of triangle 
.
be the incenter of triangle 
, and let 
, 
, and 
be arbitrary points lying on segments 
,
, and 
, respectively. The perpendicular bisectors of segments 
, 
, and 
form triangles 
. Prove that the circumcenter of triangle coincides with the circumcenter of triangle if and only if is the orthocenter of triangle 
.
