AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
with circumcircle 
is given. Let 
be the midpoint of 
, 
be the foot of altitude from 
to 
, and 
be the intersection point of 
and 
. Point 
lies on the arc of (the one that does not contain ) such that 
. Prove that 
.
be a trapezium inscribed in a circle 
with diameter . Let be the intersection point of the diagonals and 
. The circle with center 
and radius 
meets at the points 
and 
(where is on the same side of as 
). The line perpendicular to at intersects 
at 
. Prove that 
is perpendicular to 
.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
. Prove that
Equality holds when 
or 
, , and lie on a circle such that 
. The tangent line to at point intersects the tangent line to at and the line 
at and . The circle and the circumcircle of triangle 
intersect again at . Prove that 
. with orthocenter . Extend 
to meet at . The angle bisector of 
meets the midpoint of . If 
, compute the area of 
.
and . Prove that
Equality holds when or 
intersect its circumcircle at 
. The tangets to from 
determine a triangle 
, where 
is relative to and so on. Prove that lines 
are concurrent at a point 
and that belongs to the euler line of . be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at , and 
intersect 
at 
. Prove that lines 
and 
are either parallel or coincide.
be an acute triangle with circumcenter 
. Line 
intersects side at . It is known that 
and 
. Calculate the angles of the triangle.
.
and lies on 
, so 
.
and 
.
.
, then 
.
.
.
.
.
.
isoscleles,
isosceles.
is reflection of across the perpendicular bisector of then 
, and 
, wherefrom 
, i.e. 
. Further, easily, from 
, a.s.o.
and 
.
Where 
