Math and Chess Club in Ottawa

Entrance TestRules:

- Student have 90 minutes to complete the test. - For each problem, a student can earn up to 10 points with a full solution (proof).- Calculator is allowed, but using computer, tablet, or mobile phone is not allowed.- No help should be allowed from any other persons.- Students can use any paper-based books, notes.

Problem 1

Danny the Hooligan cuts the and squares out chess board. He challenges Kevin the Smart to tile the board with dominos. Can Kevin the Smart do it?

Problem 2

Is it possible to cut a square into:(a) 6 squares (not necessarily same size)(b) 7 squares (not necessarily same size)(c) 8 squares (not necessarily same size)

Problem 3

A bug sits on the vertex A of a cube. It crawls on the surface of the cube from the vertex A to the vertex B. What is the shortest path?

Problem 4

Insert the brackets into the expression below to obtain 1961:

a1 h8

(, )

4 × 8 + 5 × 6 × 7 + 9 + 2

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Math and Chess Club in Ottawa

Problem 5

The son of a professor’s father is talking to the father of the professor’s son, and the professor does not take part in the conversation. Is this possible? Give an example if you think so.

Problem 6

The figure below shows fourteen villages connected by roads.

Is there a path passing through each village once?

Problem 7

A plane can cut a cube resulting in a cross-section polygon. Below are two examples of a triangle and a square.

What is maximum number of edges a cross-section can have? Draw an example.

Problem 8

A boat goes with a constant speed downriver from the Elven Town to the Village of Traders in 3 days. It goes upriver from the Village to the Town at a possibly different but constant speed in 5 days. Bilbo Baggins took a raft to float from the Town to the Village. How long time does that take him?

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Math and Chess Club in Ottawa

Problem 9

Find the sum of the angles in the right figure below.

Problem 10

Lilian draw five identical circles as below. She coloured some parts of them with blue (b), red (r), and yellow (y). Parts with same colours are identical.

Can you find a formula to calculate a yellow (y) part based on blue (b), and red (r) parts?

∠MAN + ∠MBN + ∠MCN + ∠MDN

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