IMSO 2009 Mathematics – Taiwan Selection – Exploration

**Printed examination instructions.** Answer the following 5 questions. Write down your answer in the space provided after each question. Each question is worth 8 points. Time limit: 60 minutes.

At a recent athletics meeting, five old acquaintances, Fred, Greta, Hans, Iolo and Jan, met for the first time since leaving college, so they had much news to exchange. They lived in different towns: Acton, Buswick, Coalford, Derby and Eccles. Their different jobs were, in no particular order, engineer, lawyer, teacher, doctor and shopkeeper. Each won exactly one different event: 100 m, 400 m, 1500 m, High Jump and Javelin.

The following facts were also known:
1. Hans, the shopkeeper from Derby, won the High Jump.
2. The lawyer was from Eccles and said he was not a runner.
3. Greta was a P.E. teacher from Buswick and won the 1500 m.
4. The doctor, from Acton, did not win the 100 m.
5. The person from Derby was not an engineer.
6. Iolo was an engineer from Coalford and did not win the 400 m.
7. Jan was not a lawyer, but did win the 400 m.
8. Fred did not come from Acton and was not a runner.

(a) Which event did the person from Coalford win? (1 point)

(b) Which town did Jan come from? (1 point)

(c) What was the name of the lawyer? (2 points)

(d) Which event did the engineer win? (2 points)

(e) Which event did Fred win? (2 points)

In 2008, a school had 450 new pupils, making a total of 1600 pupils. In 2009 there were 504 new pupils, and 8% of the previous year's pupils left the school.

(a) Find the percentage increase in the number of new pupils from 2008 to 2009. (2 points)

(b) How many pupils left the school? (3 points)

(c) What was the total number of pupils in the school in 2009? (3 points)

The diagram represents a small sheet of 12 postage stamps, all perforated at their edges and all of the same value. The letters only identify the stamps.



You need 4 stamps to post a letter and want all 4 properly joined together at their edges, not merely at corners. For example, ABCD, EFGH, JKLM and FGHL work, but EFLM does not. In how many different ways can you obtain such a group of 4? Write down this number.

A domino consists of two unit squares joined edge to edge, each with a number on it. Fifteen dominoes numbered 11, 12, 13, 14, 15, 22, 23, 24, 25, 33, 34, 35, 44, 45 and 55 have been assembled into the 6 by 5 rectangle below. The individual domino boundaries have been erased. Reconstruct the dominoes by drawing the boundary lines.

Shapes can be made by joining diamonds edge to edge. Each diamond has equal sides, an angle of 60 degrees and an adjacent angle of 120 degrees, and area 1 square unit. Exactly two different shapes can be made from two diamonds; these are called bimonds, each of area 2 square units.



Each shape below is a trimond made from three edge-joined diamonds, of area 3 square units. The three pictures show the same trimond: rotations or reflections produce one from another.



(1) From a set of one diamond, two different bimonds and your collection of different trimonds, fit some pieces together into a parallelogram of area 12 square units. Draw it, showing the pieces used. (2 points)

(2) From the same set of pieces, make another parallelogram of area 12 square units but with a different perimeter. Draw it. (2 points)

(3) How many different shapes can be made from four diamonds joined edge to edge, each of area 4 square units? (4 points)