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 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
be a triangle such that 
,
and 
.Let 
be a point on side 
and 
one on side 
so that the line 
is tangent to the incircle of 
.
.
be an acute triangle. The perpendicular to passing through 
cuts the circle of diameter in 
and 
. The perpendicular to 
passing through 
cuts circle of diameter in 
and 
.,, and are cocyclic.
such that each one of them passes through the center of the other, and a square inscribed in the common part.Calculate the area of the violet square.
= 2∠AMP.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
with 
and the circle 
of radius centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of and Prove that 
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.
and be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from . with incentre 
and excentres 
and 
, opposite the vertices 
and respectively. The incircle touches 
and at 
and 
respectively. Prove that the circles 
and 
have a common point other than . and outside triangle such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect at point 
. Prove that 
.
such that for all integers 
and 
, ![\[ f(2x+f(y))+f(f(2x))=y. \]](http://latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
, consider the generating function 
. Clearly, this yields 
. Therefore, we obtain
, we have
, it is straightforward to observe that the coefficient of 
in 
is 
. This leads to the conclusion.
, consider the generating function . Clearly, this yields . Therefore, we obtain, we have, it is straightforward to observe that the coefficient of in is . This leads to the conclusion.
and 
where each summation runs over 
variables 
and 
Then find the value of 
in 
is 
,
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