# Miscellaneous Practice Problems for Regional Mathematical Olympiad

Problem 1:

Fourteen students in a class study Spanish, and eight students study French. We know that three students study both languages. How many students are there in the class if every one of them is studying at least one language?

Problem 2:

The plane is coloured using two colours. Prove that there are two identically coloured points exactly 1 metre apart.

Problem 3:

A straight line is coloured using two colours. Prove that we can find a segment of non-zero length with the endpoints and midpoint coloured the same.

Problem 4:

A 8 x 8 square is formed by 1 x 2 dominos. Prove that some pair of them forms a 2 x 2 square.

Problem 5:

We call a bus overcrowded if there are more than 50 % of the maximum allowable number of passengers inside. Children ride in several buses to a summer camp. Which is greater: the percentage of overcrowded buses or the percentage of children riding in the overcrowded buses?

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Nalin Pithwa

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