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 an isosceles trapezoid with 
. The inscribed circle 
of triangle 
meets 
at 
. Let 
be a point on the (internal) angle bisector of 
such that 
. Let the circumscribed circle of triangle 
meet line at 
and 
. Prove that the triangle 
is isosceles.
such that, for all positive integers 
and 
,
be real numbers so that 
and 
Show that
inscribed in a circle 
with center 
. Let 
the incenter of triangle and 
the contact points of the incircle with sides 
of triangle respectively . Let also 
the foot of the perpendicular line from 
to the line 
.Prove that line 
passes from the antidiametric point 
of 
in the circle .( 
is a diametre of the circle ).
be a positive integer. Two players and 
play a game in which they take turns choosing positive integers 
. The rules of the game are: takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
and 
, of equal radius intersect at different points 
and 
. Consider a circle externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
and be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
in any cell of the grid.
. and .
be non-negative real numbers such that 
Prove that
be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at . The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers 
.
+1 w
and 
be coprime integers. Call a set 
balanced if the numbers 
form an arithmetic progression with difference 
, and the numbers 
form an arithmetic progression with difference 
.
and 
, determine the maximum size of a collection of balanced sets such that every two of them have a non-empty intersection.
be an integer. The sequence 
is defined via 
, 
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer . Prove that there are finitely many pairs 
of positive integers such that 
.
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at . Prove that lies on the line .
