APMOPS 2004

A student multiplies the month and the day in which he was born by 31 and 12 respectively. The sum of the two resulting products is 170. Find the month and the date in which he was born.

Given that five whole numbers $a,b,c,d,e$ are the ages of 5 people and that $a$ is 2 times $b$, 3 times $c$, 4 times $d$ and 6 times $e$, find the smallest possible value of $a+b+c+d+e$.

Editorial convention: use positive ages, as intended by the supplied answer key.

Lines $AC$ and $BD$ meet at point $O$. Given that $OA=40$ cm, $OB=50$ cm, $OC=60$ cm and $OD=75$ cm, find the ratio of the area of triangle $AOD$ to the area of triangle $BOC$.

1000 kg of a chemical is stored in a container. The chemical is made up of 99% water and 1% oil. Some water is evaporated from the chemical until the water content is reduced to 96%. How much does the chemical weigh now? Enter the mass in kg.

A student arranges 385 identical squares to form a large rectangle without overlapping. How many ways can he make the arrangement? Note: the arrangements shown in figure (1) and figure (2) are considered the same arrangement.

A bag contains identical sized balls of different colours: 10 red, 9 white, 7 yellow, 2 blue and 1 black. Without looking into the bag, Peter takes out the balls one by one. What is the least number of balls Peter must take out to ensure that at least 3 balls have the same colour?

Find the value of $\dfrac{1\times5\times18+2\times10\times36+3\times15\times54}{1\times3\times9+2\times6\times18+3\times9\times27}$.

If the base of a triangle is increased by 10% while its height decreased by 10%, find the area of the new triangle as a percentage of the original one. Enter the numerical percentage.

A box of chocolate has gone missing from the refrigerator. The suspects have been reduced to 4 children. Only one of them is telling the truth.

John: “I did not take the chocolate.”

Wendy: “John is lying.”

Charles: “Wendy is lying.”

Sally: “Wendy took the chocolate.”

Who took the chocolate?

How many digits are there before the hundredth 9 in the following number: $9797797779777797777797777779\ldots$?

A particular month has 5 Tuesdays. The first and the last day of the month are not Tuesday. What day is the last day of the month?

In the following division, what is the sum of the first 2004 digits after the decimal point? $2004\div7=286.285714285714\ldots$

A three digit number $\overline{5ab}$ is written 99 times as $\overline{5ab5ab5ab\ldots5ab}$. The resultant number is a multiple of 91. What is the three digit number?

Which one of the following is the missing figure? Refer to the 3 by 3 pattern and choices A–F in the original image.

How many rectangles are there in the following diagram?

Placed on a table is a mathematics problem, $89+16+69+9\triangle+\square8+88$, where each of the symbols $\triangle$ and $\square$ represents a digit. Two students A and B sit on opposite sides of the table facing each other. They read the problem from their directions and both get the same answer. What is their answer?

Find the value of $\dfrac1{4\times9}+\dfrac1{9\times14}+\dfrac1{14\times19}+\cdots+\dfrac1{1999\times2004}$.

The diagram shows a right-angled triangle formed from three different coloured papers. The red and blue coloured papers are right-angled triangles with the longest sides measuring 3 cm and 5 cm respectively. The yellow paper is a square. Find the total area of the red and blue coloured papers. Enter the area in square centimetres.

The diagram shows a circular track with AB as its diameter. Betty starts walking from point A and David starts from point B. They walk toward each other along the circular track. They meet at point C which is 80 m from A the first time. Then they meet at point D which is 60 m from B the second time. What is the circumference of the circular track? Enter the distance in metres.

A triangle, figure (1), is folded along the dotted line to obtain a figure as shown in figure (2). The area of the triangle is 1.5 times that of the resulting figure. Given that the total area of the three shaded regions is 1 square unit, find the area of the original triangle. Enter the number of square units.