such that 
and 
are all perfect square.
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be the feet of the altitudes of the triangle 
,
has radius 
.
be the point such that 
,
is perpendicular to the line 
and 
similarly. Prove that 
are collinear.
such that 
Find all possible values of 
.
Y by PikaPika999
Consider the set 
of lattice points with positive coordinates in the plane. For each point 
from , we draw a segment between it and each of the points in the set ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](//latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in with a finite number of colours such that every two points joined by a segment are coloured with different colours.
Ioan Tomescu
of lattice points with positive coordinates in the plane. For each point 
from , we draw a segment between it and each of the points in the set ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](http://latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in with a finite number of colours such that every two points joined by a segment are coloured with different colours.Ioan Tomescu
