IMSO 2006 Mathematics - International Short Answer

Three signal lights flash every 12, 30 and 66 seconds respectively. They flash simultaneously at 8:30 a.m. At what time will they next flash together?

Dina has only ten-thousand and five-thousand rupiah bills. The number of ten-thousand bills is three more than twice the number of five-thousand bills. Her total is Rp355,000. How many ten-thousand bills does she have?

The principal of Makmur Jaya Elementary School is replaced every 4 years. At most how many principals will the school have from 2006 to 2020?

The shaded region has area 98 cm². Find a in centimetres. The outer rectangle is 14 cm by 9 cm; the bottom widths beside the opening are 3 cm and 4 cm.

A sequence uses only 1, 2 and 3: 1; 2,2; 3,3,3; 1,1,1,1; 2,2,2,2,2; 3,3,3,3,3,3; 1,1,1,1,1,1,1; ... . What is its 100th number?

Every whole number greater than 7 can be written as a sum of 3s, 5s, or both; for example 9=3+3+3, 10=5+5 and 19=5+5+3+3+3. Requiring every 5 to come before every 3, how many ways can 444 be expressed?

Consider numbers between 100 and 2006 made only from digits 0,1,2,3,4 with no repeated digit. How many are divisible by 6?

Two runners start together and run in opposite directions around a field of perimeter 300 m. Their speeds are 150 m/min and 125 m/min. How many times do they pass each other in the first 20 minutes?

The numbers of students in classes A and B are in ratio 1:2. Their average test scores are in ratio 8:9. Class A averages 72. Find the combined average.

The largest equilateral triangle has area 16 cm². Each new triangle joins the midpoints of the previous triangle’s sides. Continuing the shown pattern, find the area of the smallest triangle in Figure 5.

Joining every pair of vertices of a regular pentagon divides it into 11 non-overlapping regions as shown. How many regions result from doing the same to a regular hexagon?

A 20 cm × 40 cm × 80 cm wooden block is sliced into four small identical blocks. Find the largest possible sum of their surface areas.

A square and an equilateral triangle have equal perimeters. The triangle has side 8 cm. Find the square’s area.

Paint the faces of a cube so that any two faces sharing an edge have different colours. Find the minimum number of colours.

How many non-congruent triangles of perimeter 11 have integer side lengths?

Fill the eight empty cells of the shown magic square with 17,18,...,24 so that all rows, columns and both diagonals have equal sums. The middle-left cell is 25. What belongs in the starred middle-right cell?

The faces of each die have 1 to 6 dots, with opposite faces summing to 7. Four dice are arranged as shown. Faces that touch have the same number of dots. What is the total number of dots on all the touching faces?

Colour every edge of a cube red or green. To ensure every face has at least one red edge, what is the minimum number of red edges?

Different digits A,B,C satisfy AB+BA+AC+CA+BC+CB=ABC, where juxtaposition denotes a decimal number. Find the largest possible three-digit number ABC.

31513 and 34369 have equal remainders when divided by a certain three-digit number. Find the remainder.