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.
be a triangle, and let 
be on 
so that 
. Let 
intersect the circumcircle of 
again at 
. Let 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
. If 
is the circumcenter of 
, prove that 
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
, where 
and 
are not necessarily distinct. Determine 
.
be a cyclic quadrilateral with 
. Let be the midpoint of the arc 
not containing 
. Suppose there is a point 
inside such that 
and 
.
, and 
are concurrent.
. By one question, we may take three different indices 
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
be an equilateral triangle with side length 
and be a point inside the triangle such that 
. What is the area of a triangle with side lengths 
?
be the number of positive integers that divide the integer 
. For all positive integral divisors of 
, what is the sum of numbers 
?
such that the value of the
is a square of an integer.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
, , 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point is chosen on 
such taht 
and point 
is the foot of the perpendicular from to 
. Ray 
mets again at . Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
. Prove that lies on .
. 
are the midpoints of 
respectively. is the midpoint of arc 
, is symmetric to through . Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
and 
of the acute 
, outside of the triangle, the squares 
and 
are constructed. Prove that the intersection point of 
and 
lies on the height 
of .
.
.
intersects the circle 
in the point 
.
is the midpoint of 
.
and 
intersect in the point 
. satisfy the equation of 
.
