IMSO 2006 Mathematics - International Exploration

For a meeting, equilateral triangular tables are arranged in a row as shown. Only one person can sit at each open edge of a table.



(a) How many people can be seated when 20 tables are assembled in this manner?

(b) What is the minimum number of tables needed to seat 99 people in one row?

(c) What is the minimum number of tables needed to seat 100 people if there are five rows?

Three concentric circular tracks, denoted by A, B and C, are shown below. The diameter of the innermost circle is 1 unit, and the width of each track is also 1 unit. Circles with diameter 1 unit are put in the tracks so that, within each track, no circles overlap. The figure shows two non-overlapping circles in each track as an example.



(a) What is the greatest total number of circles that can fill the three tracks A, B and C?

(b) If there are six concentric circular tracks, what is the greatest number of circles that can fill the outermost track?

Positive numbers are written in order beginning with 1 as shown in the table. The number 15 appears in row $R_1$, column $C_5$.



(a) If the pattern continues, what number is in row $R_7$, column $C_7$?

(b) In which position is the number 100?

(c) What number appears directly below the number 150?

The figures below contain 5 squares in a $2\times2$ grid and 14 squares in a $3\times3$ grid.



For parts (a)–(c), you may do your working in the worksheets provided.

(a) How many squares are in the $8\times8$ grid?



(b) How many squares are in the two $8\times8$ grids with 3 coinciding units, as shown below?



(c) How many squares are in the two $8\times8$ grids with 6 coinciding units, as shown below?

Some adults and children are on one side of a river with only one boat. Everyone can row. The boat can carry no more than:
- 2 adults; or
- 2 children accompanied by 1 adult; or
- 4 children.

A crossing from one bank to the other counts as one trip.



Editorial illustration: A means adult and C means child.

(a) What is the least number of trips needed to get 5 adults and 10 children across?

(b) What is the least number needed for 10 adults and 10 children?

(c) What is the least number needed for 20 adults and 20 children?

On grid paper, polygons are drawn by connecting nearest neighboring grid points horizontally, vertically or diagonally. The length of any polygon edge is at most $\sqrt2$ units. The figure shows a triangle, quadrilaterals, pentagons and a hexagon drawn in this way.



Using the same method:

(a) Draw as many non-congruent hexagons (6-gons) as possible, other than the given example.

(b) Draw as many non-congruent heptagons (7-gons) as possible.

(c) What are the maximum and minimum possible areas of the heptagons?