[/quote]
,
and 
Prove that
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
.
.
.
to 
meets 
at points 
,
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be pairwise coprime positive integers with 
being prime and 
.![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by 
such that for all 
be real numbers such that 
Prove that
In the 
plane with the origin 
,
intersects the lines 
,
at the points 
,
be positive real numbers.![$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$](http://latex.artofproblemsolving.com/c/4/9/c4923b16be8a8faf86422477a6ed6b4b60eb2552.png)
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
and 
Prove that
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be reals such that 
. Prove that
Let 
. Prove that
Y by
Let 
be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, then
the oriented angle
is equal to the oriented angle DCB.
2.If
lies on the segment 
, then
the oriented angle
plus the oriented angle 
equals 
.
3.If the oriented angle
plus the oriented angle 
equals 0°, then
lines
and are parallel.
4.If the oriented angle plus the oriented angle equals , then
lines and are parallel.
be four distinct points in the plane.Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, thenthe oriented angle
is equal to the oriented angle DCB.2.If
lies on the segment 
, thenthe oriented angle
plus the oriented angle 
equals 
.3.If the oriented angle
plus the oriented angle 
equals 0°, thenlines
and are parallel.4.If the oriented angle
plus the oriented angle equals , thenlines
and are parallel.
