Here is a Olympiad Problem and i have a solution for it already , please tell me know if i will get full marks for my solution or not (i think my solution is short than official solution)? You can also post your alternative solutions ^_^
Let x=0.$a_1$$a_2$$a3$... where $a_i$ is 1 if nos. of positive divisors of $i$ is even and 0 if they are odd .Prove that x is irrational .
My solution is : $a_i$ has odd nos of divisors iff i is a perfect square else it has even nos of divisors . Thus , $a_i$ is 0 iff $i$ is perfect square else it is 1. Also a nos is rational if in its decimal representation , there is periodicity in it. But difference/gap between two perfect square goes on increasing , ie gap between two 0s after decimal goes on increasing and thus periodicity is not possible . Hence x is irrational .