In rugby it is possible to score 3, 5 or 7 points at a time. If South Africa wins the next Rugby World Cup and scores 34 points in the final, the highest number of times 3 points could have been scored by South Africa is
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10
In the school library there are $1\,024$ books of which $64$ are detective stories. Twenty-four of these detective stories are about Sherlock Holmes. What is the probability that a book you selected randomly is a detective story which is not about Sherlock Holmes?
Let $P$ be the product of any three consecutive positive odd integers. What is the highest common factor (greatest common divisor) of all such numbers $P$?
At a height of $h$ metres above sea level, the distance $d$, in kilometres, to the visible horizon is given approximately by the formula $d = 8\sqrt{\frac{h}{5}}$. A mountaineer on the side of Table Mountain can see a distance of $80$ kilometres to the horizon. How high, in metres, is the mountaineer above sea level?
Mrs. Habana turns $45$ years old in $2016$ and her son $17$. In which year will Mrs. Habana turn double her son's age?
(A) $2033$ (B) $2044$ (C) $2031$ (D) $2040$ (E) $2027$
A certain pen can draw a line $0.4$ mm wide and $1$ km long before it runs out of ink. The largest square you can colour in with this pen has an area, in square metres, of
(A) $4\,000$ (B) $400$ (C) $40$ (D) $4$ (E) $0.4$
In $\triangle RUG$, $RU = 10$ cm, $UG = 21$ cm, $RG = 17$ cm and $RY$ is perpendicular to $UG$ at point $Y$. The length of $RY$, in cm, is
(A) 8 (B) 5 (C) 7 (D) 6 (E) 8.1
What is the minimum number of integers which must be removed from the first 50 positive even integers so that the sum of the remaining integers is $2016$?
A sequence of $2016$ terms is constructed as follows: The first two terms of the sequence are both equal to $3$. Starting from the third term, each subsequent term is the sum of the preceding two terms. Each of the terms of this sequence $3$, $3$, $6$, $9$, $...$ is now divided by $2$ and the remainders are added. What is the sum of all the remainders of the $2016$ terms?
(A) $504$ (B) $1\,008$ (C) $1\,344$ (D) $1\,512$ (E) $2\,016$
A box contains $5$ red pens, $6$ blue pens and $4$ green pens. Three pens are consecutively taken from the box, at random and without replacement. The probability that the first pen is red, the second pen blue and the third pen green, is
(A) $\frac{7}{225}$ (B) $\frac{1}{5}$ (C) $\frac{9}{82}$ (D) $\frac{4}{91}$ (E) $\frac{3}{14}$
$D$ is a point on side $AB$ of $\triangle ABC$, $E$ is a point on $CD$ and $F$ is a point on $CE$. The areas of triangles $AED$, $AEC$, $BFD$ and $BFC$ are $6$, $10$, $17$ and $7$, respectively. What is the area of $\triangle BEF$?
If $169! = 1 \times 2 \times 3 \times \cdots \times 169$ is written as the product of prime numbers, how many times would $13$ appear as a factor?
(A) 12 (B) 13 (C) 14 (D) 15 (E) 16
$A$ and $B$ are opposite vertices of a regular hexagon. $C$ and $D$ are midpoints of two opposite sides. If the area of the hexagon is $126$, then $AB \times CD$ is
(A) $129$ (B) $132$ (C) $84\sqrt{3}$ (D) $168$ (E) $248$
How many grams of pure gold must be added to five grams of a mixture that consists of 10% gold and 90% of some other metal so that the mixture will contain 20% gold?
(A) 0.125 (B) 0.25 (C) 0.375 (D) 0.5 (E) 0.625
Let $n$ and $k$ be integers satisfying $n \ge 2$ and $k \ge \frac{5}{2}n - 1$. Prove that in every choice of $k$ distinct points among all integer points $(x, y)$ with $1 \le x, y \le n$, there exists a circle going through at least four distinct chosen points.
