APMOPS 2005

There is a triangle and a circle on a plane. Find the greatest number of regions that the plane can be divided into by the triangle and the circle.

The diagram shows a rectangle $ABCD$. $P$ and $Q$ are the midpoints of $AB$ and $CD$ respectively. What fraction of the rectangle is shaded?

Source issue: no shaded region is visible in the supplied image. The original is preserved below; the intended region cannot be uniquely identified.

There is a glass of water and a glass of wine. A small amount of water is poured from the glass containing water into the glass containing wine. Then an equal amount of the wine-water mixture is poured back into the glass containing water. Which one of the following statements is correct?

Two squares, with lengths 4 cm and 6 cm respectively, are partially overlapped as shown in the diagram below. What is the difference between shaded area A and shaded area B? Enter the area in square centimetres.

The diagram shows a dartboard. What is the least number of throws needed in order to get a score of exactly 100?

Find the value of $1-\dfrac56+\dfrac7{12}-\dfrac9{20}+\dfrac{11}{30}-\dfrac{13}{42}+\dfrac{15}{56}-\dfrac{17}{72}+\dfrac{19}{90}$.

Julia and Timothy each has a sum of money. Julia’s amount is $3/5$ that of Timothy’s. If Timothy were to give Julia 168 dollars, then his remaining amount will be $7/9$ that of Julia’s. How much does Julia have originally? Enter the amount in dollars.

When a number (the dividend) is divided by another number (the divisor), the quotient is 4 and the remainder is 8. Given that the sum of the dividend, the divisor, the quotient and the remainder is 415, find the dividend. Note: when 17 is divided by 3, the quotient is 5 and the remainder is 2.

Six different points are marked on each of two parallel lines. Find the number of different triangles which may be formed using 3 of the 12 points.

How long, in hours, after 12 noon, will it take for the hour hand of a clock to lie directly over its minute hand for the first time?

The lengths of two rectangles are 11 m and 19 m respectively. Given that their total area is 321 m², find the difference in their areas. Note: both their widths are whole numbers. Enter the area in square metres.

Editorial convention: each stated length is the longer side, so its width does not exceed it; this is needed for the supplied answer.

Lala speaks the truth only on Monday, Wednesday, Friday and Sunday. Nana speaks the truth only on Monday, Tuesday, Wednesday and Thursday. Find the day when both said “Yesterday I lied”.

The diagram shows a semicircle $ACB$ with diameter $AB=21$ cm. Angle $DAB=30^\circ$ and arc $DB$ is part of another circle with centre $A$. Find the perimeter of the shaded region, using $\pi=22/7$. Enter the perimeter in centimetres.

Find the value of $20042005\times20052004-20042004\times20052005$.

In how many ways can $7/12$ be written as a sum of two fractions in lowest term given that the denominators of the two fractions are different and are each not more than 12?

There are 72 students in Grade 6. Each of them paid an equal amount of money for their mathematics text books. The total amount collected is $\square35.0\square$ dollars, where two of the digits indicated with $\square$ cannot be recognized. How much money did each student pay for the books? Enter the amount in dollars.

Numbers such as 1001, 23432, 897798, 3456543 are known as palindromes. If all of the digits 2, 7, 0 and 4 are used and each digit cannot be used more than twice, find the number of different palindromes that can be formed.

There are three classes A, B and C. Class A has 2 more students than class B. Class B has 1 more student than class C. The product of the numbers of students in the three classes is 99360. How many students are there in class A?

Find the value of $1+2+2^2+2^3+\cdots+2^{11}$. Note: $2^2=2\times2$, $2^3=2\times2\times2$.

Three boys Abel, Ben and Chris participated in a 100 m race. When Abel crossed the 100 m mark, Ben was at 90 m. When Ben crossed the 100 m mark, Chris was at 90 m. By how many metres did Abel beat Chris? Note: they were running at constant speeds throughout the race.