be a fixed acute-angled triangle. Consider some points 
and 
,
. Let the perpendicular bisector of 
,
intersect the lines 
, respectively. We call the pair 
,
is cyclic.
and 
are interesting. Prove that 
and 
be four prime numbers such that 
Prove that the sum of the four prime numbers is divisible by 
.
and 
and one starting number from the set 
used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number 
is reached or exceeded.
Show that Bob cannot win.
for which 
and 
are both perfect squares.
denotes the sum of the positive integer divisors of 
Let 
be the least positive integer such that 
is divisible by 
for all positive integers 
be the set of integers. Find all functions 
such that ![\[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\]](http://latex.artofproblemsolving.com/9/f/d/9fdb11dbf102dfe0783b1d31466e276420e3913d.png)
for all 
with 
.
and let 
with the internal bisector of the angle 
.
be the circumcenter of triangle 
and 
the orthocenter of triangle 
.
is equal to the circumradius of triangle 
Y by Adventure10, Mango247
is a cube (with 
and 
faces, and 
edges). 
is a line which intersects or is parallel to the lines 
and 
. meets the line 
at 
(which may be the point at infinity). Let 
. The plane 
meets the line 
at 
. Show that 
. The plane 
meets the line 
at 
.Show that
.Hence or otherwise show how to construct the point
at the intersection of and the plane .Find the distance between
and the line 
and the distance between and the line in terms of 
.Find a relation between these two distances that does not depend on
.Find the locus of
.Let
be the envelope of the line as varies. Find the intersection of with the faces of the cube.
