[/quote]
,
and 
connect the points 
and 
These arcs lie in the same half-plane defined by line 
in such a way that arc 
lies between the arcs 
and 
Point 
lies on the segment 
Let 
,
be three rays starting at 
lying in the same half-plane, 
being between 
and 
For 
denote by 
the point of intersection of 
and 
(see the Figure below). Denote by 
the curved quadrilateral, whose sides are the segments 
and arcs 
and 
We say that this quadrilateral is 
if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals 
are circumscribed, then the curved quadrilateral 
is circumscribed, too.
Prove that
be integers, let 
be a set with 
elements, and let 
be pairwise distinct non-empty, not necessary disjoint subset of 
is called nice if there exists an index 
such that ![\[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\]](http://latex.artofproblemsolving.com/3/d/a/3da5ad1bcade98ee08caa35763de226027bcb2c6.png)
Prove that the number of nice functions is at least 
.
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
Prove that
are fixed on a line in this order. Let 
be a circle passing through 
the intersection of the tangents to 
at 
.
and the line 
Prove that
Where 
be a triangle with circumcentre 
.
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
.
be the circle passing through 
satisfying
Prove
.
has at most 
be a cyclic quadrilateral with 
and 
.
such that 
.
and 
are selected on segment 
such that 
.
is a cyclic quadrilateral.
Y by Adventure10
Let 
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations
![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](//latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](http://latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
and 
.
,
and 
.
and we need to maximize this for 
.
,
.![\[ \rho = \frac {\cos{\frac {\pi}{6}}}{\sin{\frac {\pi}{2}}} = \frac {\sqrt {3}}{2}
\]](http://latex.artofproblemsolving.com/5/e/0/5e049296f93c7894fe4c7c6e64d99319ef7e1e11.png)
and the answer is 
.
,
.
.
.
,
and 
are 
since
is an increasing function over (0, 90), 
is also increasing function over (60, 90).
maximizes at 
=> 
![\[ a^2+y^2=b^2+x^2=2(ax+by)\]](http://latex.artofproblemsolving.com/2/0/e/20ea4850a93a01e6458bd0f2eddc90dde91ef752.png)
,
.
,
.
satisfies the equation. Hence 
,
be maximized at 
?
which matches the other solutions...
,
.
on ![$[0,30]$](http://latex.artofproblemsolving.com/5/c/0/5c0d28e4a7273d9c9719062979757fc6b3272d08.png)
.
for all 
in this range. Then: ![\[\cos x\le k\sin(x+60)=k\cdot\left(\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x\right)\]](http://latex.artofproblemsolving.com/f/b/8/fb8160a01b7f2675f19e4fe0610994ad2e940ce9.png)
![\[\rightarrow 0\le \left(\frac{\sqrt{3}}{2}-1\right)\cos x+\frac{k}{2}\sin x\rightarrow 0\le (\sqrt{3}k-2)\cos x+k\sin x\]](http://latex.artofproblemsolving.com/c/7/1/c7184c5cb46816cf5e223f36b2fc194db706e665.png)
![\[\rightarrow (2-\sqrt{3}k)\cos x\le k\sin x\rightarrow \frac{2-\sqrt{3}k}{k}\le \tan x,\]](http://latex.artofproblemsolving.com/e/a/8/ea8d072fb7f1f0fa8bfc9566694c32c308a26fe3.png)
for all 
,
must be less than or equal to 
,
.
.
attains a maximum of 
on the interval.
for 
,
and 
in that order (clockwise). The problem statement gives that these three points form an equilateral triangle. Let 
be the line parallel to the x-axis going through 
be the line parallel to the y-axis going through 
.
has length 
and width 
,
maximized reduces to finding the triangle such that 
giving us an answer of 
.
