IMSO 2008 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.

The diagram shows a network of water pipes, with eight taps, each shown by a symbol ×, at positions A, B, C, D, E, F, G and H. Water enters along three pipes at the left, indicated by →, and cannot pass a tap that is “off”. Each tap is either “on” or “off”. How many choices of taps in the “on” position result in water coming out of the pipe at the extreme right, assuming no more than three taps can be “on” at the same time?

The diagram shows a 3 × 3 × 3 frame with 27 holes in which marbles can be placed. A–B–C, L–M–N and X–Y–Z are some possible lines of three marbles.



(a) From A, how many different lines of three marbles are possible? (3 points)

(b) What is the total number of different possible lines of three marbles in this frame? (5 points)

Triominoes is played with a set of tiles, all sides of which are equal in length. Each vertex of every tile is marked with one of the digits from 0 to 5. Different vertices on the same tile may have the same digits, so the two examples below are both included.



All possible tiles, including those with duplicate or triplicate digits, are in the set. How many possible tiles are there? The following mirror-image tiles are different tiles.

At the end of the season, the total prize money in a football tipping competition is shared among the five place-getters under these rules:

(i) All prizes are different positive whole numbers of dollars.

(ii) First prize is the sum of second and third prizes.

(iii) Second prize is the sum of fourth and fifth prizes.

(iv) The higher the placing, the larger the prize. There are no ties.

(1) Find the smallest possible total prize money. (3 points)

(2) If the total prize money is 72 dollars, find all possible distributions among the five place-getters. (5 points)

An n-dragon is a set of n consecutive positive integers. The first two-thirds form the tail; the remaining one-third form the head. The sum of the numbers in the tail equals the sum in the head. For example, the nine consecutive integers 2,3,4,5,6,7,8,9,10 form a 9-dragon:

- Its tail is 2,3,4,5,6,7: six numbers with sum 27.
- Its head is 8,9,10: three numbers with sum 27.

(1) Find a 21-dragon. (3 points)

(2) Find the sum of the tail of a 99999-dragon. (5 points)