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 cyclic quadrilateral with 
. Let 
be the midpoint of the arc 
not containing 
. Suppose there is a point 
inside such that 
and 
.
, and 
are concurrent.
circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or 
. must be odd.
. 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 ., so that the equation 
has two real roots, so that the one is the square of the other.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
be a convex pentagon such that 
. Suppose that the midpoint of is the circumcenter of triangle 
. Let 
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
be a cyclic pentagon inscribed in a circle of centre which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter .
and 
are the incenter and the -excenter of triangle , respectively. be the midpoint of arc 
on the circumcircle, and be the foot of the to . The line 
intersects the circumcircle again at 
. For
on the circumcircle of triangle , let 
intersect at 
. Prove
are concyclic.
is . The incenter of the triangle is , and the intouch triangle is 
. Let 
be the orthocenter of 
. Prove that , , and are collinear.
,then the triangles 
and 
are similar homotetic with centered at 
( with parallel sides ).On the other hand 
are orthocenter and circumcenter of the triangle , 
are orthocenter and circumcenter of the triangle , so 
are collinear.
