[/quote]
,
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
be a triangle. Let 
be a points on the segment 
.
be the mid-point of 
intersect 
in 
in 
.
,
and 
e
are tangent to 
and 
respectively. 
intersects line 
.
.
intersects 
.
.
that satisfy the following condition for any real numbers 
and 
such that:
Prove that
.
be the midpoints of 
and 
respectively and let 
be the projection of 
.
on 
.
intersect 
at points 
and 
.
intersct 
.
intersect 
at 
and 
intersect 
.
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
are chosen on 
,
is the foot of the altitude from 
to 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
and 
and 
.
and 
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are 
for every positive integer 
prime numbers, find for which values of 
terms of the sequence.
such that, for all real numbers 
with 
,
s
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
Y by
Let 
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
