Cool Number Theory
by Fermat_Fanatic108, Mar 19, 2025, 1:41 PM
For an integer with 5 digits 
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
Prove that
.
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.Prove that
.
,
and 
e
are tangent to 
at 
and 
respectively. 
intersects line 
.
intersects 
,
intersects 
.
intersects 
.
.
Prove that
that satisfy the following condition for any real numbers 
and 
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 
respectively. Let the circle with centre 
passing through 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
.
be the midpoints of 
and 
respectively and let 
be the projection of 
.
.
intersect 
at points 
and 
.
intersct 
.
intersect 
at 
and 
intersect 
.
are concurrent.
such that:
be a points on the segment 
.
.
intersect 
and 
respectively. Determine 
.
such that, for all real numbers 
with 
,
May 2018
April 2018