APMOPS 2002

How many numbers are there in the following number sequence? $1.11,1.12,1.13,\ldots,9.98,9.99$.

What is the missing number in the following number sequence? $\frac12,\frac1{12},\frac1{30},\frac1{56},\boxed{\phantom{000}},\frac1{132}$.

Observe the pattern and find the value of a. The three diagrams have (top, left, bottom, right) entries (3,4,5,35), (4,5,6,54), and (5,6,7,a).

Find the value of $\frac1{2002}+\frac2{2002}+\frac3{2002}+\cdots+\frac{2000}{2002}+\frac{2001}{2002}$.

The average of 10 consecutive odd numbers is 100. What is the greatest number among the 10 numbers?

What fraction of the figure is shaded, when each side of the triangle is divided into 3 equal parts by the points?

The figure is made up of two squares of sides 5 cm and 4 cm respectively. Find the shaded area.

Find the area of the shaded figure. The grid consists of 1 cm by 1 cm squares.

Draw a straight line through the point A to divide the 9 circles into two parts of equal areas.

In the figure, $AB=AC=AD$, $\angle ABC=70^\circ$ and $\angle ADC=50^\circ$. Find $\angle BAD$.

In the sum, each box represents a non-zero digit. What is the sum of all the 6 missing digits? The column addition is $202+\boxed{\phantom0}\boxed{\phantom0}\boxed{\phantom0}+\boxed{\phantom0}\boxed{\phantom0}\boxed{\phantom0}=2002$.

The average of n whole numbers is 80. One of the numbers is 100. After removing the number 100, the average of the remaining numbers is 78. Find n.

The list price of an article is \$6000. If it is sold at half price, the profit is 25%. At what price must it be sold so that the profit will be 50%?

$\frac17$ of a group of pupils score A for Mathematics; $\frac13$ score B; $\frac12$ score C; and the rest score D. If a total of 100 pupils score A or B, how many pupils score D?

At 8.00 a.m., car A leaves Town P and travels along an expressway. After some time, car B leaves Town P and travels along the same expressway. The two cars meet at 9.00 a.m. If the ratio of A’s speed to B’s speed is 4:5, what time does B leave Town P?

Which one of the following is the missing figure? The figure pattern and options A, B, C and D are shown.

A rectangle is folded along a diagonal as shown. The area of the resulting figure is $\frac58$ of the area of the original rectangle. If the area of the shaded triangle is 18 cm², find the area of the original rectangle.

The square ABCD is made up of 4 triangles and 2 smaller squares. Find the total area of square ABCD. The four marked side segments are each 10 cm.

The diagram shows two squares A and B inside a bigger square. Find the ratio of the area of A to the area of B.

There are 3 straight lines and 2 circles on the plane. They divide the plane into regions. Find the greatest possible number of regions.