## part 3 of 3 — Solutions to Pre RMO Oct 2014

**Question Set A.**

**Question 20. **

What is the number of ordered pairs where A and B are subsets of such that neither nor

?

**Solution. **Just list down A and B explicitly. Note that A and B are disjoint and that is an ordered pair.

**Question 13. **For how many natural numbers n between 1 and 2014 (both inclusive) is an integer?

**Solution. **Firstly, note that is even. Hence, n is odd.

Also, to yield an exact integer, so .

Now, do the one of the core tricks for problem solving in number theory. Plug and play with numbers 🙂

Put . This works.

Next, note that for some positive integer k. Hence, we get

From this we observe that is the only possible solution. Hence, the answer is 1.

**Note : Question 16: HW to be posted on the blog 🙂**

More later,

Nalin Pithwa

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