I am currently solving ACM Timus 2058 , a problem about palindromes.

In palindromic tree every node represents a distinct palindrome (lets say , p) in a string and this node links to another node which represents the longest proper suffix of it(p) which is a palindrome. Eg. If a node u represents "aabaa" then it's link will be a node v which represents "aa", since "aa" is longest palindromic suffix of "aabaa".

When solving that problem, I observed an interesting property (and I could not prove it's correctness) about palindrome.

 [  Note : If X is a string , then here by (X^n) I mean concatenation of n X string.  If X="ab" then X^3= "ababab".  And XY represent concatenation of X and Y . X="a" ,Y="b" then XY="ab" ] 

Property-1: " If a palindrome P has longest palindromic suffix Q and if | length of Q | >= (| length of P | / 2) then palindrome P has a structure like (X^n)Y where n>=2 , X and Y are string ( | Y | < | X | and Y could be a null string ). Here, X will be a prefix of P and length of X= |P|-|Q|. "

Eg-1 : If X='aba' and Y=''(NULL) then 'abaaba' = (X^2), 'abaabaaba' = (X^3), 'abaabaabaaba = (X^4)' etc If X='bba' and Y='bb' then 'bbabbabb' = [(X^2)Y], 'bbabbabbabb' = [(X^3)Y] , 'bbabbabbabbabb' = [(X^4)Y] etc

EG (of property): P="bbabbabbabb" and longest suffix palindrome of it is Q="bbabbabb" then by property X="bba". Now P can be represent as XXXY, where Y="bb" .

And then Y would be a palindrome too.

But everything depends on Property-1. I did some paper works and found it happens. But I think it is not enough to find the correctness of the property.

Edit: This property is correct, as symbols on positions that differs by |P| - |Q| are equal ( Thanks Um_nik for providing the proof).

Why symbols on positions that differs by |P|-|Q| would be equal? You can find the proof here

[ Sorry For my bad English. I tried my best to explain ,though I am not good at explaining. If you don't understand any part of it , you can ask me ]