[/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 
.
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 Mango247
Given an non-isosceles acute-angled triangle 
. On rays 
and 
, respectively points 
and 
such that the quadrangle 
is cyclic. Point 
is the foot of the sltitude drawn from vertex 
on side 
. Prove that if 
then is the center of the circumcircle of the triangle 
.
. On rays 
and 
, respectively points 
and 
such that the quadrangle 
is cyclic. Point 
is the foot of the sltitude drawn from vertex 
on side 
. Prove that if 
then is the center of the circumcircle of the triangle 
.This post has been edited 1 time. Last edited by parmenides51, Jan 6, 2021, 8:21 AM
and lies on line 
such that 
and it is the unique intersection of those lines. By hypothesis 
by cyclicity of 
.
and we are done
