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.
for which, for all 
, the following identity holds:![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
such that 
. Find all such 
.
be a convex quadrilateral with 
, 
and 
. Denote by 
and 
the reflections of 
in lines 
and 
, respectively. Suppose that the segments 
and 
meet the line 
at 
and 
, respectively. Prove that the circumcircles of triangles 
and 
are tangent to each other.
. The tournament adopts an elimination system, and the winner advances to the next round. What is the probability that the Argentine team will meet the Brazilian team in the entire tournament?. 
. 
. 
. 
. 
be a triangle with centroid 
. Points 
and 
are chosen on rays 
and 
, respectively, such that![\[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\]](http://latex.artofproblemsolving.com/0/a/d/0ad01e29194358ea0d3e10cb74b0769d1bd3a656.png)
Prove that 
.
and 
for which![\[ \left \lfloor \frac{a^2}{b} \right \rfloor + \left \lfloor \frac{b^2}{a} \right \rfloor = \left \lfloor \frac{a^2 + b^2}{ab} \right \rfloor + ab.\]](http://latex.artofproblemsolving.com/c/3/6/c3682397d21443cb720e56251c260b984392a84a.png)
be a triangle with 
, incenter 
, circumcenter 
, and circumcircle 
. The line 
meets 
at , and meets again at 
. Let 
be the reflection of over 
. Let the line through perpendicular to meet at 
. 
meets 
and 
at and , respectively. Suppose that the circumcircle of 
meets again at 
. Prove that 
, , 
, are concurrent if and only if the sides of 
form an arithmetic progression.
such that 
for all 
.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
with 
right and 
, 
. On the ray 
we take two points and 
so that 
and 
. Consider the squares 
and 
containing inside and , respectively. If 
and 
, 
and 
, prove that 
and that , and are collinear. inside a square we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
. Proof that![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
the angles and 
are equal and the bisector of 
passes through the midpoint of the side . If it is known that 
, calculate 
.
and 
so 
and 