Some problems for Pre-RMO practice

Below are a few problems, which I think might boost a student’s confidence for Pre-RMO.

1) Factorize (show a proof also): a^{3}+b^{3}+c^{3}-3abc

2) Bad calibration.

My bathroom scale is set incorrectly but otherwise it works fine. It shows 10 kilograms when Dan stands on it, 14 kilograms when Sarah stands on it, but 22.5 kilograms Dan and Sarah are both on it. Is the scale set too high?

3) Getting inside a brick.

You  have 3 identical rectangular bricks and a ruler. Must you use a formula, such as the Pythagorean theorem, to find the length of the brick’s diagonal?

4) In the running.

In a cross-country run, Sven placed exactly in the middle among all participants. Dan placed lower, in tenth place, and Lars placed sixteenth. Is it possible to figure out how many runners took part in the race?

5) The sands of time.

Maia bought two unusual sandglasses. One measures a nine-minute interval, and the other measures a thirteen-minute interval. A certain love potion needs to boil for exactly thirty minutes. is it possible to measure such a time interval with these sandglasses under the additional stipulation that you turn over the glass(es) for the first time just as the potion starts to boil?

6) Anyone for tennis?

Each child in a school plays either soccer or tennis. One-seventh of the soccer players also play tennis, and one ninth of the tennis players play soccer. Do more than half the children play tennis?

7) Tiresome parrot.

We recently bought a parrot. The first day it said “O” and the second day “OK”. The third day the parrot said “OKKO” and the day after that “OKKOKOOK”. If this doubling pattern continues and the parrot squawks every second, will it get through squawking on the sixteenth day?

8) A wire cube.

A 12 inch long wire is to be divided into a number of parts and from these we want to construct the frame (that is, the edges) of a cube of 1 inch on a side. Can you do it with three pieces?

9) Paper folding.

You have a strip of paper that is two thirds of a meter long. However, you need a strip exactly half a meter long. Must you have a ruler to cut off such a length?

10) No one twice as rich.

One hundred children in a school are counting their money. Each child has between 1 and 100 cents, and no child has the same amount as any other child. Is it possible to divide the children into two groups so that no child in either group will have twice as much money as any other child in the same group?

Good luck 🙂 Happy problem solving…keep on trucking…

More later…

Nalin Pithwa

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