1980 Canadian MO Problems

Problem 1

If is the decimal expansion of a number in base , such that it is divisible by , determine .

Solution

Problem 2

The numbers from to are printed on cards. The cards are shuffled and then laid out face up in rows of cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrrangement, do the cards in the rows still increase from left to right?

Solution

Problem 3

A triangle is given with one constant angle and a constant inradius. Explicitly define all triangles for which the minimum perimeter is achieved.

Solution

Problem 4

A gambling student tosses a fair coin. She gains point for each head that turns up, and gains points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] points is .

Solution

Problem 5

A parallelepiped has the property that all cross sections, which are parallel to any fixed face , have the same perimeter as . Determine whether or not any other poyhedron has this property.

Solution

Resources

1980 Canadian MO