APMOPS 2001

Find the value of $0.1+0.11+0.111+\cdots+0.1111111111$.

Find the missing number in the box: $5\times\boxed{\phantom{000}}\div3\times4-299=2001$.

Find the missing number in the following number sequence: $1,4,10,22,46,\underline{\qquad},190,\ldots$.

If numbers are arranged in 3 rows A, B and C according to the following table, which row will contain the number 1000?

A: 1, 6, 7, 12, 13, 18, 19, ...

B: 2, 5, 8, 11, 14, 17, 20, ...

C: 3, 4, 9, 10, 15, 16, 21, ...

How many 5-digit numbers are multiples of 5 and 8?

John started from a point A, walked 10 m forwards and then turned $36^\circ$ right. Again he walked 10 m forwards and then turned $36^\circ$ right. He continued walking in this manner and finally returned to the starting point A. How many metres did he walk altogether?

What fraction of the figure is shaded?

How many triangles are there in the figure?

Between 12 o’clock and 1 o’clock, at what time will the hour hand and minute hand make an angle of $110^\circ$?

The rectangle ABCD of perimeter 68 cm can be divided into 7 identical rectangles as shown in the diagram. Find the area of the rectangle ABCD.

Find the smallest number such that

(i) it leaves a remainder 2 when divided by 3;

(ii) it leaves a remainder 3 when divided by 5;

(iii) it leaves a remainder 5 when divided by 7.

The sum of two numbers is 168. The sum of $\frac18$ of the smaller number and $\frac34$ of the greater number is 76. Find the difference between the two numbers.

There are 325 pupils in a school choir at first. If the number of boys increases by 25 and the number of girls decreases by 5%, the number of pupils in the choir will become 341. How many boys are there in the choir at first?

Mr Tan drove from Town A to Town B at a constant speed of 40 km/h. He then drove back from Town B to Town A at a constant speed of 70 km/h. The total time taken for the whole journey was 5.5 h. Find the distance between the two towns.

Which one of the following is the missing figure? Use the transformation shown by the first pair of figures. Options A, B, C and D are shown below.

Which two of the following solid figures can be fitted together to form a cuboid? The figures are labelled A, B, C and D.

In how many different ways can you walk from A to B in the direction $\uparrow$ or $\rightarrow$, without passing through P and Q?

In the figure, ABCD is a square and EFGC is a rectangle. The area of the rectangle is 24 cm². Given that $AE=\frac58 AB$, find the length of one side of the square.

The diagram shows a circle and 2 quarter circles in a square. Find the area of the shaded region. (Take $\pi=3.14$.) The square has side 20 cm.

The area of rectangle ABCD is 24 cm². The areas of triangles ABE and ADF are 4 cm² and 9 cm² respectively. Find the area of triangle AEF.