APMOPS 2014

The figure shows a regular octagon. Find the value of angle $SRT$ in degrees, where $R$ is the centre of the star.

Which of the following statement is correct?

(1) $\frac{15}{19}<\frac79<\frac34$

(2) $\frac34<\frac{15}{19}<\frac79$

(3) $\frac79<\frac{15}{19}<\frac34$

(4) $\frac34<\frac79<\frac{15}{19}$

(5) $\frac79<\frac34<\frac{15}{19}$

Enter the option number.

Find the smallest positive integer with sum of digits equal to 29.

Abel and Jim run a race together on a 300 metres track. They start simultaneously at the same point. Abel runs at a constant speed of 5 m/s, while Jim runs at a constant speed of 4.2 m/s. How many full laps has Abel run before he is able to overtake Jim for the first time?

Only one of the following four numbers is a perfect square. Which one is it?

(1) 76186; (2) 750235; (3) 921438; (4) 2660161.

Enter the option number.

As shown in the diagram, $\triangle ABC$ is an isosceles right-angle triangle with $AB=28$ cm. $BC$ is the diameter of the semi-circle and point $D$ is the midpoint of arc $BC$. Find the area of the shaded region in cm$^2$. (Take $\pi$ to be $\frac{22}{7}$.)

Three chess pieces, each of the colour red, black and white, are to be placed on a $7\times7$ chessboard. If any two of the three pieces cannot be placed in the same row or the same column, how many ways are there to place the three chess pieces?

There are 6 bags containing 18, 19, 21, 23, 25 and 34 balls respectively. One bag contains only red balls while the other five bags contain only blue balls. Jason takes three bags and Jamie takes two bags. The remaining bag contain red balls. It is known that now Jason has twice as many blue balls as Jamie does. Find the number of red balls in the remaining bag.

A water tank can be filled by using tap A for 8 hours followed by tap B for 15 hours. The same tank can also be filled by using tap A for 5 hours followed by tap B for 24 hours. How long will it take (in hours) to fill the tank by using tap A only?

A square piece of paper is folded along the diagonals twice, as shown in the diagrams below. Then a cut is made along the dotted line. When the paper is unfolded, which one of the following diagrams shows how the paper appears? Enter the option number.

Find the value of $20142013\times20132014-20132013\times20142014$.

Alan, Ben, Chris and Daniel took part in a science quiz. The quiz consisted of 5 True or False questions. The table below shows the answers given by each of them and their scores. Each correct answer was given 1 point.

Answers to Questions 1–5 and score:

Alan: T, F, F, F, F; score 2.

Ben: F, T, F, F, T; score 3.

Chris: F, T, F, T, T; score 4.

Daniel: F, T, T, F, T; score unknown.

How many points did Daniel score?

A computer program generated all five-letter code words that can be formed by using letters A, B, C, D, E, F, G, X, Y, Z (repetition is allowed). When all these words are sorted according to the alphabetical order, the following list is obtained: AAAAA, AAAAB, ..., AAAAZ, AAABA, AAABB, ..., ZZZZY, ZZZZZ. Find the number of code words between CZYGB and XEFDA, not including these two code words.

Find the largest 3-digit integer $N$ such that when $N$ is divided by 3, 7, 11, the remainders are 1, 3 and 8 respectively.

Find the value of $10\times\left(\frac{1}{1\times2}+\frac{5}{2\times3}+\frac{11}{3\times4}+\cdots+\frac{71}{8\times9}+\frac{89}{9\times10}\right)$.

As shown in the diagram, a square is divided into 9 regions. Region $E$ is a square and the other regions are rectangles. If the area of rectangles $A,B,C$ are $18$ cm$^2$, $63$ cm$^2$ and $189$ cm$^2$ respectively, find the perimeter of rectangle $D$.

The sum of 47 distinct positive integers is 2014. If there are $n$ even numbers among the 47 integers, what is the least possible value of $n$?

A 9-digit integer $\overline{abcdefghi}$ is formed using digits $1,2,3,4,5,6,7,8,9$ without repetition. If $A=\overline{abc}+\overline{bcd}+\overline{cde}+\overline{def}+\overline{efg}+\overline{fgh}+\overline{ghi}$, find the largest possible value of $A$.

A semi-circle with diameter $AG$ is shown in the diagram, the entire arc of the semi-circle is divided into 6 equal parts by points $B,C,D,E,F$. $DF$ and $CG$ are straight lines. Given that the area of the semi-circle is $60$ cm$^2$, find the area of the shaded region in cm$^2$.

As shown in the diagram below, cubes with side length of 1 cm are placed together to form a sequence of solids. Find the surface area of the 20th solid in cm$^2$.