[/quote]
,
be positive real numbers . 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 
.
be a point on the arc 
(not containing 
and 
.
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
be real numbers such that 
Prove that
Let 
Prove that
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 an equilateral triangle, 
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)
.
such that 
and 
for any real numbers 
.
be real numbers satisfying![\[x^2 +y^2+z^2 = 1 \]](http://latex.artofproblemsolving.com/d/1/8/d18e31dce3f312d786cca935d497743ba76091ef.png)
Prove that![\[ Q = xy + yz +2xz \le \dfrac{1+\sqrt{3}}{2}\]](http://latex.artofproblemsolving.com/e/5/6/e563c1223d410c64506a55e1ced878df96b7971c.png)
be non-negative real numbers.Prove that :![\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]](http://latex.artofproblemsolving.com/6/2/3/623d50afffd81b2783ff98568791164b53747567.png)
Y by
Consider acute 
with 
, 
and area 
. The circle 
with diameter 
meets 
at 
. Let the orthocenter of the triangle be 
, connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
with 
, 
and area 
. The circle 
with diameter 
meets 
at 
. Let the orthocenter of the triangle be 
, connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
,
.
intersects the line 
in the point 
.
in the point 
.
.
.
.
intersects the circle 
in the points 
and 
.
.
.
is a rectangle, we have 
,
.
,
.
by orthocenter reflection, the radius for both circles is 
.
,
as 
and the projection of 
.
.
meet 
,
.
,
.
,
.
.
