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 real numbers so that 
and 
Show 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 .
, s.t :
by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](http://latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
. We define the sequences 
and 
by 
, and 
, 
for 
. Show that there exists a positive integer 
such that ![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
is marked on side of parallelogram 
, and on the side 
- point 
so that the circumscribed circle of 
is tangent to line segment 
. Prove that the circumcircle of triangle 
is tangent to line 
.
be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross at . On circle , let and 
be the mid-arc points of and 
, respectively. Let 
cross 
at and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
. be a triangle with 
-mixtilinear incircle 
, and let the tangents from 
and 
to different from 
and 
meet the circumcircle again at 
and 
, and let 
and 
meet again at and . Show that the -excircle of 
is tangent to the circumcircle.
and a point 
outside the circle such that 
. The perpendicular bisector of 
at 
and 
.The segment 
intersects 
at . The perpendicular bisector of
intersects 
at and 
.The rays 
intersect at 
respectively. Prove that the triangles 
and 
are equilateral triangles and the lines , 
are concurrent.
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
of positive integers satsifying 
.
penguin and each one is numbered using numbers 
. penguin 
is called 
if 
where 
is 
prime divisor of (
(where k is number of 's prime divisors). what is maximum ratio of 
where 
is number of in antarctica?(
)
and let be the midpoint of the side .
and 
. Find the measure of 
.
, s.t :
We have
Law of sines in 
is equivalent to 
