[/quote]
,
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
less than 
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
,
where 
is the number of times the number 
appears in the list. So, for example, for the list 
,
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of 
,
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist 
(use in a problem for junior) to practice. Can anyone share to me?
,
be midpoints of 
and 
be reflections of 
.
have the same centroid.
and 
are acute and 
.
,
.
.
,
,
.
at 
.
at 
.
be the midpoint of 
,
meet 
.
at 
is tangent to 
be positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be the maximal number of the vertices of a convex lattice polygon with exactly 
for 
.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
Y by Adventure10
The angles of the triangle 
are less than 
.
The point
inside the triangle is such that 
.
Let
be the intersection points of the medians and let 
be the intersection points of the altitudes in the triangles 
respectively.
Prove the equality
,
where
is the intersection point of the medians in the triangle .
(M. Kurylo, Lypova Dolyna)
are less than 
.The point
inside the triangle is such that 
.Let
be the intersection points of the medians and let 
be the intersection points of the altitudes in the triangles 
respectively.Prove the equality
,where
is the intersection point of the medians in the triangle .(M. Kurylo, Lypova Dolyna)
