APMOPS 2013

Find the last 5 digit of the sum.

$$127354+27354+7354+354+54+4.$$

Find the sum of all two-digit numbers whose units digit and tens digit are both even.

The diagram shows a semi-circle with centre $O$ overlapped with a parallelogram $ABCD$. The diameter, $AB$, of the semi-circle is $12$ cm. Find the total area of the shaded regions in $\mathrm{cm}^2$.

Abel, Ben and Charlie took part in SMOPS 2012, which comprised 30 questions. They had correctly answered 26, 23 and 18 questions respectively. What is the least possible number of questions that were answered correctly by all 3 students?

Find the value of $555\times554555-554\times555554$.

The diagram shows a cube with side length $5$ cm. If a rectangular tunnel with dimensions $2$ cm by $3$ cm is made in the middle of the cube, find the amount of increase in the total surface area of the resulting solid in $\mathrm{cm}^2$.

The average value of four whole numbers $\overline{a}$, $\overline{b5}$, $\overline{c17}$ and $\overline{d432}$, where $a,b,c$ and $d$ each represents the first digit of a number, is $1735$. Find the value of $a+b+c+d$.

The diagram shows a regular octahedron, which is a solid composed of eight equilateral triangles.

Which of the following patterns can be folded into a regular octahedron?

The sum of 2 prime numbers is equal to $2013$. Find the product of these two numbers.

The figure shows a big square, which is divided into four identical rectangles and one small square. Given that the area of the small square is $16\ \mathrm{cm}^2$, the area of each rectangle is $140\ \mathrm{cm}^2$, find the width of each rectangle in cm.

A 5-digit number written in the form $\overline{24abc}$ has the last three digits unknown. If this number is divisible by $3,4$ and $5$ respectively, find the greatest possible value that $\overline{abc}$ can take.

In the diagram, an ant is moving from $A$ to $B$. If the ant is only allowed to move to the right or upwards along the grid lines, how many different paths are there from $A$ to $B$?

The diagram shows a parallelogram $ABCD$. $E$ is the midpoint of $AD$. $F$ is the midpoint of $EC$. If the area of the triangle $BFD$ is $9\ \mathrm{cm}^2$, find the area of the parallelogram $ABCD$ in $\mathrm{cm}^2$.

If we write $\dfrac{2013}{1990}$ in the form

$$a+\frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}}},$$

where $a,b,c,d$ and $e$ are positive integers, what is the value of $a+b+c+d+e$?

In the given diagram, find the value (in degrees) of

$$\angle A+\angle B+\angle C+\angle D+\angle E+\angle F.$$

In how many different ways can four children share $8$ identical chocolates so that each child get at least one?

In the given diagram, each circle contains a natural number and the diagram satisfies the following conditions:

1. The number labelled along each edge represents the difference between the numbers in the two circles joined by the edge.
2. The sum of the numbers in the $5$ circles is equal to $1979$.

Find the number in circle $A$.

A certain type of water bottle is sold at \$10 in both Store A and Store B. Mrs. Lim would like to buy a few water bottles for a Children's Home. Store A sells the water bottle with an offer of “Buy 5 Get 1 Free” (no free bottles for buying fewer than 5 water bottles); store B gives a 15% discount for customers who buy 4 or more water bottles. What is the least amount of money (in \$) that Mrs. Lim needs to spend in order to get 14 water bottles?

A square of side length $18$ cm is inscribed in a circle. Semi-circle are constructed on its sides, as shown in the diagram. Find the total area of the four shaded lunes in $\mathrm{cm}^2$.

Four teams participated in a soccer tournament. Each team played against all other teams once each. $3$ points were awarded for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament, the four teams have obtained $5,1,x,6$ points respectively. Find the value of $x$.