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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 .

Maggi Iggam
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    I like it! I'm not sure if your solution here is an outline or the full proof that you are going to submit. For submission, you should rigorously justify the remarks that you have made here. – Euler....IS_ALIVE Feb 03 '13 at 01:50
  • Thanks :) This is the full proof i would have submitted! I am much weak in explaining in written but here i cant see what more remarks i can make ? – Maggi Iggam Feb 03 '13 at 01:55
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    @MaggiIggam: You would need to justify the remarks about the number of divisors, and you would also need to justify that the gap between any two perfect squares is strictly increasing. No matter how trivial these may seem, they're part of a full solution. – Clayton Feb 03 '13 at 02:07
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    There are some minor imprecisions. If a number is rational, then its decimal representation is ultimately periodic. (Your version states the result in the other direction, true but not what we need here.) Also, what you intend to say is that the gap between consecutive squares is increasing. It is a good argument overall. However, it might be a very early question in the contest, and those are often marked in a quite picky way. The imprecisions may well bring you down substantially. – André Nicolas Feb 03 '13 at 02:16
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    @Clayton: I am not so sure about justifying, for example, the remark about the number of divisors. A contest kid knows this sort of thing. As long as the quoted result is stated correctly, one would not necessarily expect a proof. – André Nicolas Feb 03 '13 at 02:38
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    @AndréNicolas: I've not participated in too many contests ($2$ I think), so I wasn't sure what would be typical to know versus not being rigorous enough. Thanks for the tip! – Clayton Feb 03 '13 at 02:57
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    My inclination would be to award $7,8$, or $9$ marks out of $10$, depending on the standard of the competition, or $4$ out of $5$. However, apart from taking the Putnam four times back in the 1960s, I was never involved much with contests, so my inclination doesn’t mean much. – Brian M. Scott Feb 03 '13 at 03:10
  • Thanks for the tips everyone , vote up to all ^^ Proffesor Brian , yeah this was contest of stage 2 to select top 35 students of India and your marks do mean coz you are still much inclined to maths ;) Clayton : Thanks i will surely keep in mind to prove small trivial and obvious reasonings too ^^ Thanks @AndréNicolas but do i need to prove things like difference between 2 perfect squares is strictly increasing ? – Maggi Iggam Feb 03 '13 at 19:34
  • Every competition has its own inner rules. You really need to talk to appropriate people to find out about the local expectations. My answer was based on grading APMO, CMO, and a number of lower level contests. – André Nicolas Feb 03 '13 at 19:41
  • Yeah you are sure correct @AndréNicolas :) – Maggi Iggam Feb 03 '13 at 20:44

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