APMOPS 2003

What is the greatest possible number of right angles formed by 4 straight lines in a plane?

What is the next number in the following sequence: 123, 347, 7815, 151631?

The following figure is made up of 13 squares. On the answer sheet provided, show two ways to draw a straight line to divide the figure into two parts of equal areas.

There are 30 children in a class. 12 of them can play badminton. 8 of them can play table-tennis. Find the greatest possible number of children who can play neither game.

Find the value of $\frac12+\frac13+\frac23+\frac14+\frac24+\frac34+\cdots+\frac1{100}+\frac2{100}+\frac3{100}+\cdots+\frac{99}{100}$.

A card, with the starting position $\uparrow$, is first rotated clockwise through $3000^\circ$, then anticlockwise through $2745^\circ$, and finally clockwise through $1320^\circ$. Which option indicates the final position? A: up; B: down; C: down-left; D: down-right; E: right. Fill in A, B, C, D or E.

What fraction of the following figure is shaded?

There are three closed boxes containing a red ball, a green ball and a yellow ball respectively. They are all labelled incorrectly. David opens the box labelled red and finds a green ball. What is the colour of the ball in the box labelled green?

A four-digit number 7a4b is a multiple of 18. Find digits a and b if the four-digit number is to be as small as possible.

Given that $2^1=2$, $2^2=2\times2$, $2^3=2\times2\times2$, $2^4=2\times2\times2\times2$, and so on, find the remainder when $2^{2003}$ is divided by 3.

A water plant grew so fast that it doubled the area of water surface it covered every minute. It was brought to a 100 m by 50 m pond at 11.00 a.m. At 12 noon it covered the entire pond. Find the time when it covered a quarter of the pond.

There are 10 points on the 5 sides of a pentagon as shown. How many triangles can be drawn with any three of the points as vertices?

Each big bus has 39 seats and costs 100 dollars to rent. Each small bus has 30 seats and costs 80 dollars to rent. 267 men need to travel by bus. Given that every man must have a seat and no seat is left empty, what is the total rental cost?

AF, BG, CH, DI and EJ are straight lines. What is the sum of angles A, B, C, D, E, F, G, H, I and J?

In triangle ABC, CX = 2BX and CY = 3AY. The area of triangle AXY is 20 m². Find the area of triangle ABC.

There are 20 questions in a mathematics test. 7 marks are awarded for each correct answer, 4 marks deducted for each incorrect answer, and no marks given for each unanswered question. If a pupil scores 100 marks, how many questions were left unanswered?

In the figure, the area of rectangle ABCD is 36 cm². P, Q and R are the midpoints of AB, BC and CD respectively. S is a point on AD. Find the total area of the shaded parts.

Mrs Lim was carrying a basket of eggs to the market when a passer-by bumped into her. She dropped the basket and all the eggs broke. The passer-by, wishing to pay for the loss, asked, “How many eggs were there in your basket?” Mrs Lim replied, “I do not remember exactly, but when I count by 3s, there are 2 eggs left over; when I count by 5s, there are 3 left over; and when I count by 7s, there are 4 left over.” At least how many eggs were in the basket?

When multiplying two 2-digit numbers, Sam miswrote the ones digit of one number as 0 instead of 8 and obtained 190. June miswrote the ones digit of the same number as 6 and obtained 304. Find the correct product.

A rectangular block is cut into 12 identical cubes as shown. The total surface area of the 12 individual cubes is 160 cm² more than the surface area of the original block. What is the volume of the original block?