be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
,
,
.
is the midpoint of side 
of an acute—angled triangle 
.
is symmetric to the orthocenter 
inside triangle 
.
.
is given. Suppose that 
,
and 
are different positive integer numbers such that 
.
.
and 
Prove that
be real positive number satisfy that: 
.
,
integers and 
satisfy: 
are divisible by k. Prove that:
is divisible by k
such that 
and
.
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
Y by Adventure10, Mango247
We are given a ruler with two marks at a distance 1. With its help we can do all possible constructions as with a ruler with no measurements, including one more: If there is a line 
and point 
on , then we can construct points 
for which 
. By using this ruler, construct a perpendicular from a given point to a given line.
and point 
on , then we can construct points 
for which 
. By using this ruler, construct a perpendicular from a given point to a given line.This post has been edited 1 time. Last edited by Pinko, Sep 19, 2019, 8:04 AM
