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 pentagon with 
and 
. Let 
and 
be points on 
and 
, respectively, such that 
is cyclic. Let 
be the midpoint of 
. Prove that lines 
, 
, and 
concur.
be real numbers such that 
and 
and determine the equality cases.
be the set of real numbers. Determine all functions 
that satisfy the equation![\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]](http://latex.artofproblemsolving.com/6/b/2/6b2acc0a1d76866e461fdf5d96126806a9ead8dd.png)
for all real numbers 
and 
.
over 
is linear
with incenter 
. Line 
and 
intersects 
and 
at 
and 
, respectively. Let and 
be the midpoints of and . Line 
meets the external bisector of angle 
at 
and line 
meets the external bisector of angle 
at 
. Prove that 
are concylic.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
, is the midpoint of minor arc 
, 
is the circumcenter of 
and 
is its diameter. The line parallel to 
through meets segment at 
, and extended at . Lines 
and 
meet at , lines 
and meet at . Prove that 
.
of positive integers such that
such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
+1 w
cut out 
congruent regular 
gons from a sheet of paper and placed these gons on the table such that some parts of each of these gons may be covered by others. We say that a vertex of one of the afore-mentioned gons is 
if it is not in the interior of another gon that is placed on top of it. For any 
determine the minimum possible number of visible vertices.
inscribed in a circle, 
. On the side 
, point is marked so that 
. Prove that the perpendicular bisector of the segment 
bisects the arc that does not contain point 
.
inscribed in a circle, . On the side , point is marked so that . Prove that the perpendicular bisector of the segment bisects the arc that does not contain point .
and 
Prove that
