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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