APMOPS 2011

If the following figure is folded into the shape of a cube, what is the number opposite to the face marked $P$?

Find the value of $(1-\frac12)(1-\frac13)(1-\frac14)\cdots(1-\frac1{2010})(1-\frac1{2011})$.

A circle of radius 1 m has some points lying on its circumference. Find the minimum number of points such that at least two points are less than 1 m apart.

Three sides of a four-sided figure are of lengths 4 cm, 9 cm and 14 cm respectively. If the largest possible length of the fourth side is $x$ cm where $x$ is a whole number, find the value of $x$.

Find the largest prime number that divides $(1\times2\times3\times\cdots\times97\times98)+(1\times2\times3\times\cdots\times98\times99\times100)$.

$ABCD$ is a square and $BCE$ is an equilateral triangle. If $BC$ is 8 cm, find the radius of the circle passing through $A,E,D$ in cm.

Find the value of $\frac{19}{20}+\frac{1919}{2020}+\frac{191919}{202020}+\cdots+\frac{\underbrace{1919\cdots19}_{2011\text{ copies of }19}}{\underbrace{2020\cdots20}_{2011\text{ copies of }20}}$.

The following diagram shows a circle of radius 8 cm with the centre $R$. Two smaller circles with centres $P$ and $Q$ touch the circle with centre $R$ and each other as shown in the diagram. Find the perimeter of triangle $PQR$ in cm.

Peter and Jane competed in a 5000 m race. Peter’s speed was 4 times that of Jane’s. Jane ran from the beginning to the end, whereas Peter stopped running every now and then. When Jane crossed the finish line, Peter was 100 m behind. Jane ran a total of $x$ m during the time Peter was not running. Find $x$.

If numbers are arranged in three rows $A,B,C$ in the following manner, 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, ...

The following diagram shows 5 identical circles. How many different straight cuts are there so that the five shaded circles can be divided into two parts of equal areas?

A test with a maximum mark of 10 was administered to a class. Some results are shown below. The average mark of those scoring more than 3 is 7, while the average mark of those getting below 8 is 5. Given that none scored zero, find the number of pupils in the class.

Scores 1, 2, 3, ..., 8, 9, 10 have respectively 1, 3, 6, ..., 4, 6, 3 pupils.

A test comprises 10 true or false questions. Find the least number of answer scripts required to ensure that there are at least 2 scripts with identical answers to all the 10 questions.

$P_n$ is defined as the product of the digits in the whole number $n$. For example, $P_{19}=1\times9=9$, $P_{32}=3\times2=6$. Find $P_{10}+P_{11}+\cdots+P_{98}+P_{99}$.

$ABC$ is a triangle with $BC=8$ cm. $D$ and $E$ lie on $AB$ such that the vertical distance between $D$ and $E$ is 4 cm. Find the area of the shaded region $CDE$ in $\text{cm}^2$.

The points $A,B,C,D,E,F$ are on the two straight lines as shown. How many triangles can there be formed with any 3 of the 6 points as vertices?

A group of 50 girls were interviewed to find out how many books they had borrowed from the school library in April. The total number of books borrowed by the girls in April was 88, and 18 girls had borrowed only 1 book each. If each girl borrowed either 1, 2 or 3 books, find the number of girls that had borrowed 2 books each.

The following diagram shows a triangle $PQR$ on a 2 by 7 rectangular grid. Find the sum of angle $PQR$ and angle $PRQ$ in degrees.

The following diagram shows a rectangle where the areas of the shaded regions are $5\text{ cm}^2$, $6\text{ cm}^2$, $10\text{ cm}^2$, $54\text{ cm}^2$ and $x\text{ cm}^2$ respectively. Find $x$.

The following diagram shows two squares $ABCD$ and $DGFE$. The side $CD$ touches the side $DG$. If the area of $DEFG$ is $80\text{ cm}^2$, find the area of triangle $BGE$ in $\text{cm}^2$.