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

Question Set A.

Question 20.

What is the number of ordered pairs $(A,B)$ where A and B are subsets of ${1,2,3,4,5}$ such that neither $A \subseteq B$ nor

$B \subseteq A$?

Solution. Just list down A and B explicitly. Note that A and B are disjoint and that $(A,B)$ is an ordered pair.

Question 13. For how many natural numbers n between 1 and 2014 (both inclusive) is $\frac {8n}{9999-n}$ an integer?

Solution. Firstly, note that $9999-n$ is even. Hence, n is odd.

Also, $8n \geq 9999-n$ to yield an exact integer, so $n \geq 1111$.

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

Put $n=1111$. This works.

Next, note that $(9999-n)k=8n$ for some positive integer k. Hence, we get

$9999k=(k+8)n$ From this we observe that $n=1111$ 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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