APMOPS 2016

Find the value of $2016+2015+2014-2013-2012-2011+2010+2009+2008-2007-2006-2005+\cdots+6+5+4-3-2-1$.

In the figure below, each of the three lines bisected the figure into two regions of equal areas. What is the relationship between shaded area $A$ and shaded area $B$?

(A) $A>B$; (B) $A<B$; (C) $A=B$; (D) Insufficient information to deduce.

Enter the option letter.

Find the number of fractions, among those listed below, which are in their simplest form: $\frac16,\frac27,\frac38,\ldots,\frac{2010}{2015},\frac{2011}{2016}$.

It is given that $a\oplus b=\frac{2}{a^2}+\frac1b$, which of the following equations is correct?

(A) $2\oplus4=4\oplus2$; (B) $3\oplus6=6\oplus3$; (C) $4\oplus8=8\oplus4$; (D) $1008\oplus2016=2016\oplus1008$.

Enter the option letter.

In the diagram below, the equilateral triangle has a length of 3 cm and the hexagon has a length of 1 cm. Given that the ratio of the area of the equilateral triangle to the area of the hexagon is $a:2$, find the value of $a$.

The diagram shows an isosceles triangle $ABC$ in which $\angle BAC=90^\circ$ and $BC=6$ cm. $F$ is a point on $BA$ produced such that $FD$ is perpendicular to $BC$ and it intersects $AC$ at $E$. $D$ lies on $BC$. Find the total length of $DE$ and $DF$.

Daniel drives his car in the following manner:

He drives along for 20 m then turns left;
Then drives along for 20 m then turns right;
Then drives along for 20 m then turns right;
Then drives along for 20 m then turns right;
And he repeats the same cycle.

The diagram below shows the path that Daniel drove. From which point (1, 2, 3 or 4) did he begin his journey?

Three cubes have different lengths of 3 cm, 4 cm and 5 cm respectively. All faces of the cubes were painted red. If all three cubes were cut up into smaller 1-cm cubes, find the number of 1-cm cubes that has at least one face painted red.

In the diagram below, each side of the equilateral triangle $ABC$ is divided into six equal segments. Given that every small triangle in the diagram has an area of 1 cm$^2$, find the area of the quadrilateral $DEFG$ in cm$^2$.

Two rabbits, Harry and Bunny, leave from points A and B respectively and travel towards each other. Harry’s speed is 1.5 times of Bunny’s speed. If the two rabbits first meet at a point that is 12 m away from the mid-point of AB, find the distance of AB.

It is known that the area of three different sides of a rectangular cuboid is in a ratio of 2 : 3 : 5. The total length of all its edges is 124 cm. Find the volume of the rectangular cuboid.

There is a sequence of the first 300 positive integers 1, 2, 3, ..., 299, 300. If the numbers that are divisible by 5 or 7 are removed from the sequence, find the 123rd number in the remaining sequence.

2016 distinct points are chosen within triangle $ABC$ such that no three points among the 2019 points (inclusive of $A,B,C$) lie on a straight line. Using these 2019 points as vertices, how many smaller triangles can triangle $ABC$ be cut into?

Find the value of $84\times\left(\frac{1}{1\times3}-\frac{2}{3\times5}+\frac{3}{5\times7}-\frac{4}{7\times9}+\cdots+\frac{9}{17\times19}-\frac{10}{19\times21}\right)$.

Given that $a,b,c$ are prime numbers, not necessarily distinct, and their product is the sum of 7 consecutive positive integers. Find the smallest possible value of $a+b+c$.

Find the number of zeroes at the end of the resultant product $1\times4\times7\times10\times\cdots\times697\times700$.

In the figure below, not drawn to scale, the area of triangle $ABC$ is 1 cm$^2$. The points $D,E,F$ are on $BC,CA,AB$ produced such that $BD=2BC$, $CE=3CA$ and $AF=4AB$. Find the area of triangle $DEF$.

Six 10-cent coins have the same height as five 20-cent coins when stacked on top of each other. Four 10-cent coins have the same height as three 50-cent coins when stacked on top of each other. Using 10-cent, 20-cent and 50-cent coins to stack into three stacks of equal height, a total of 124 coins have been used. What is the total value of these coins? Enter the value in dollars.

What is the remainder when $3^{2016}+2$ is divided by 11?

In the figure below, find the value of $\angle A+\angle B+\angle C+\angle D+\angle E+\angle F+\angle G$.