[/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 points on the segment 
such that 
.
be the mid-point of 
intersect 
and 
in 
respectively. Determine 
.
,
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 
.
.
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 Rounak_iitr
Let 
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
are cyclic.
and 
are cyclic.
a desired
and 
are similar
and 
are also similar, proving the equality.
and 
then 
