[/quote]
,
,
be a point on the side 
and 
.
and 
are the feet of the altitudes from 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
such that for every 
,
,
equals the number of times 
be the sequence 
so that 
for each 
and so that the decimal number 
is divisible by 
for each 
.
is irrational.
points on the plane, no three of which are collinear, and some set of line segments between them. We say that a subset of these line segments is called a "pairing" if every one of these 
,
distinct (but not necessarily disjoint) pairings.
be a polynomial with real coefficients such that 
for all 
.
for some real constant 
and 
.
where 
.
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does 
.![\[ 2^n -2, \;\; 3^n -3, \;\; 4^n -4, \;\; 5^n-5, \, \ldots \ldots \]](http://latex.artofproblemsolving.com/d/b/1/db17ef301bf3fd01f8b310f91a14b9dee045e11f.png)
is 
.
?
be a function such that 
.
.
as its circumcentre, 
as its nine-point centre.
,
and 
to be the midpoint of the smaller arcs.
as the isogonal conjugate of the Nagel point (i.e. the exsimillicenter of the incircle and circumcircle)
as the ninja point of 
as the ninja point of the contact triangle
Points 
are collinear, that is 
Points 
and 
.![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
such that, for all positive integers 
and 
,
be a positive real number. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
and 
Prove that
Y by Adventure10, Mango247
On the sides 
, 
, 
of an acute-angled triangle 
, we erect (outwardly) the squares 
, 
, 
, respectively. On the sides 
and 
of the triangles 
and 
, we erect (outwardly) the squares 
and 
.
Prove that
.
Comment. This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more:
.
Darij
, 
, 
of an acute-angled triangle 
, we erect (outwardly) the squares 
, 
, 
, respectively. On the sides 
and 
of the triangles 
and 
, we erect (outwardly) the squares 
and 
.Prove that
.Comment. This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more:
.Darij
be the simmetrical point of 
wrt 
and similarly 
of 
,
and 
be the translations of point 
of the vectors 
,
