[/quote]
,
for which there exists a sequence 
of positive integers satisfying the following conditions:[list]
for all integers 
for all integers 
the sequence 
is a permutation of
[/list]
be distinct real numbers such that 
Find the value of 
Find the value of 
Find the value of 
.
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
or 
.
and 
and the lines 
and 
meet at 
.
and 
Prove that
and 
,
outside triangle 
and 
.
and 
,
.
intersect 
.
for which each cell of 
table can be filled with one of the letters 
and 
,
and one third are 
to 
with 
.
,
diagonals of two types. A diagonal of first type consists all cells 
is a constant, and the diagonal of this second type consists all cells 
is constant.
.
Let 
.![$$ \frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$](http://latex.artofproblemsolving.com/f/9/1/f9119e071c12778037616c624c7b9ca75356e8e3.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{2\sqrt[3]{9}}$$](http://latex.artofproblemsolving.com/2/b/2/2b2d8a011229fe88a81cb75f9b018ad3a6245faa.png)
![$$ \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$](http://latex.artofproblemsolving.com/5/f/6/5f673cd494b1909ca237db63e96f949c24b6f2ad.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{\sqrt[3]{3}}{5}$$](http://latex.artofproblemsolving.com/b/c/3/bc343a07c88dbd98ebffdfb78f311b7b82683d65.png)
.
Let 
.
Y by
1. Point 
is selected on segment 
, then on segments 
and 
are constructed equilateral triangles 
and 
, lying in one half-plane relative to the straight line . Points 
divide the segments 
in a ratio of 
, respectively. Prove that points 
form an equilateral triangle.
2. On the sides and of triangle 
, isosceles triangles and with the same angles at top. Points divide segments , respectively in the same ratio 
. Prove that triangle 
is similar to triangle .
3. Two similar triangles
and 
are given, and their orientation coincides (the order of traversing the corresponding the vertices are the same). Points 
are the midpoints of segments 
, 
, 
respectively. Prove that triangle is similar to the first two.
4. Let (in the notation of the previous section) the lines containing the altitudes
and 
of the first two triangles intersect at the point 
, and the points 
and 
lie, respectively on the segments 
and 
. It turned out that these triangles are visible from points at the same angles. Prove that 
.
is selected on segment 
, then on segments 
and 
are constructed equilateral triangles 
and 
, lying in one half-plane relative to the straight line . Points 
divide the segments 
in a ratio of 
, respectively. Prove that points 
form an equilateral triangle.2. On the sides
and of triangle 
, isosceles triangles and with the same angles at top. Points divide segments , respectively in the same ratio 
. Prove that triangle 
is similar to triangle .3. Two similar triangles
and 
are given, and their orientation coincides (the order of traversing the corresponding the vertices are the same). Points 
are the midpoints of segments 
, 
, 
respectively. Prove that triangle is similar to the first two.4. Let (in the notation of the previous section) the lines containing the altitudes
and 
of the first two triangles intersect at the point 
, and the points 
and 
lie, respectively on the segments 
and 
. It turned out that these triangles are visible from points at the same angles. Prove that 
.
