[/quote]
,
![\[ 2^n -2, \;\; 3^n -3, \;\; 4^n -4, \;\; 5^n-5, \, \ldots \ldots \]](http://latex.artofproblemsolving.com/d/b/1/db17ef301bf3fd01f8b310f91a14b9dee045e11f.png)
is 
.
?
be a function such that 
.
.
,
as its circumcentre, 
as its orthocentre and 
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
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the 
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
.
and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
.
and 
,
,
and 
intersects on 
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
larger than 
,
,
)![\[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]](http://latex.artofproblemsolving.com/0/8/f/08fc55e0cc886c4c44f678e75eb07e4d814c4a77.png)
Y by StefanS, Adventure10, Mango247
Let 
and 
be three circles with centers 
and 
respectively, such that no center is inside of the other two circles. Circles 
and 
intersect at 
and 
, circles and intersect and 
and , circles and intersect at 
and . Let 
be a point on such that the line 
intersects at 
and the line 
intersects at 
, such that is nor inside nor inside and is nor inside nor inside .
a) Prove that
is simular to 
b) Prove that the
. Is it possible to reach equation?
P$ denotes the area of a triangle*
and 
be three circles with centers 
and 
respectively, such that no center is inside of the other two circles. Circles 
and 
intersect at 
and 
, circles and intersect and 
and , circles and intersect at 
and . Let 
be a point on such that the line 
intersects at 
and the line 
intersects at 
, such that is nor inside nor inside and is nor inside nor inside .a) Prove that
is simular to 
b) Prove that the
. Is it possible to reach equation?
P$ denotes the area of a triangle*
and 
are supplement, so are 
and 
.
and 
,
Draw a line through 
to intersect 
and 
;
which parallels 
intersects 
.
and 
are similar because of equal angles. However, 
(I let you prove this) and ![$[DEF]$](http://latex.artofproblemsolving.com/c/5/b/c5ba7afa12298272441c17f763cea35fd1402bf3.png)
= 
×![$[O_1O_2O_3]$](http://latex.artofproblemsolving.com/7/9/a/79a27f625ccb188c80b88c418359323dc81fd3c7.png)
is reached (sic) when 
