[/quote]
,
.
and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
are both on 
and 
,
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)
,
,
,
respectively. Let 
.
lies on 
.
such that, for all positive integers 
and 
,
be an integer, and let 
and 
be sequences of non-negative integers. For each integer 
,
and 
,
is the integer such that 
is divisible by 
equals the following arithmetic mean: ![\[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\]](http://latex.artofproblemsolving.com/1/1/7/117ca4963c8e443fd3cbfb55a907bee4be295151.png)
Suppose that neither 
nor 
of the numbers 
are zero.
is given on the plane. Let the points 
and 
)
are taken such that 
is an equilateral hexagon. Suppose that the circumcenters of 
are coincident. Then determine the least possible value of:
Note: 
area of 
such that ![\[f(x)(f(yf(x)+1))=f(x)+f(y)\]](http://latex.artofproblemsolving.com/1/0/6/1069c29ad9eeb7fe8597d7bac806006a12ff50c9.png)
for all positive reals 
and 
and also satisfies the property that ![\[\mathbb{R}^+\subseteq\frac{\text{Im}(f)}{\text{Im}(f)},\]](http://latex.artofproblemsolving.com/1/0/2/102f23eabefae24b158fbd7516b58b1e879c1b74.png)
or in other words, the set of positive reals is a subset of the set ![\[\left\{\frac{x}{y}\mid x,y\in\text{Im}(f)\right\}\]](http://latex.artofproblemsolving.com/f/d/4/fd4e9cf0bfc63603529ded7679e2811560219338.png)
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
intersect 
are concyclic.
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 
.![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
Y by
Points 
and 
are marked on the sides 
and 
of the triangle 
respectively. Segments 
and 
intersect at point 
, and segments 
and 
intersect at point 
. The circumcircles of the triangles 
and 
intersect the side 
at points 
and 
respectively. Lines 
and 
intersect at point 
. Prove that the lines 
and 
are either parallel or concurrent.
(A.Voidelevich)
and 
are marked on the sides 
and 
of the triangle 
respectively. Segments 
and 
intersect at point 
, and segments 
and 
intersect at point 
. The circumcircles of the triangles 
and 
intersect the side 
at points 
and 
respectively. Lines 
and 
intersect at point 
. Prove that the lines 
and 
are either parallel or concurrent.(A.Voidelevich)
