[/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 
side length and with center 
.
is drawn where point 
.
,
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 a non-isosceles triangle with orthocenter 
,
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, Mango247
Let 
be a quadrilateral inscribed in the circle 
. For a point 
, its projections 
on the lines 
, respectively, are considered. Prove that if 
is the orthocentre of the triangle 
for some point 
, different from 
, then this holds for every point of the circle.
be a quadrilateral inscribed in the circle 
. For a point 
, its projections 
on the lines 
, respectively, are considered. Prove that if 
is the orthocentre of the triangle 
for some point 
, different from 
, then this holds for every point of the circle.
be the projections of 
and 
are cyclic
is a diamter
is a diameters.
and 
are diameter 
