APMOPS 2017

What is the smallest 4-digit number in the following sequence? $5,14,29,50,77,\ldots$

60% of the employees in Team A are males while 60% of the employees in Team B are females. There are 360 more employees in Team A than in Team B. If the number of female employees in Team A and Team B are equal, what is the total number of employees in the two teams?

Two fifty-cent coins are placed next to each other on a table top, forming an angle $\theta$ as shown in the figure below. Each coin takes the shape of a dodecagon with 12 sides of equal length. Find the value of $\theta$ in degrees.

Source note: the intended answer additionally uses regular dodecagons; equal side lengths alone do not specify all angles.

What is the remainder when $6\times2017^{2018}$ is divided by 11?

There were 4 questions in a mathematics test and the total number of marks was not 100. The number of marks allocated to question 1 was the highest, followed by questions 2, 3 and 4 (lowest). The number of marks allocated to question 1 was 22 and question 3 was 7 marks higher than question 4. Student A answered questions 1, 2 and 3 correctly and obtained 54 marks. Student B answered questions 2 and 4 correctly and obtained $x$ marks. Find $x$.

Alvin has many apples. He gives $\frac13$ of his apples plus $\frac23$ of an apple to Bob. He then gives $\frac14$ of his remaining apples plus half an apple to Chris. Afterwards, he gives half of his remaining apples to David. Lastly, he gives half of his remaining apples plus half an apple to Ed. In the end, Alvin has 5 apples left. How many apples does Alvin have initially?

The diagram below shows four circles $c_1,c_2,c_3,c_4$ with diameters $AB,AC,CD,DB$ respectively. If the circumference of $c_1$ is 2017, find the sum of the circumferences of all four circles.

Find the value of $217\times216216217-216\times217217216$.

In the star figure below, $\angle p+\angle q+\angle r+\angle s+\angle t=600^\circ$. Find $\angle A+\angle B+\angle C+\angle D+\angle E$ in degrees.

Given that

$$\frac AB=2-\cfrac1{2-\cfrac1{2-\cfrac1{\ddots-\cfrac1{2-\frac{2016}{2017}}}}}.$$

has 2017 layers as shown, where $A,B$ are positive integers, find $A+B$. Note that

$$2-\cfrac1{2-\frac{2016}{2017}}$$

is the first layer.

Source note: no condition that $A/B$ is in lowest terms is printed; this affects uniqueness.

A number $P$ has 9 factors whereas another number $Q$ has 10 factors. If the lowest common multiple of $P$ and $Q$ is 2800, find $Q$.

Mrs Lee bought $\frac23$ as many pens as pencils. Each pen cost \$1.20 more than each pencil. The total cost of the pens was \$12.60 more than the total cost of the pencils. Given that Mrs Lee spent a total of \$66.60 on all the pens and pencils, find the number of pencils she bought.

Given that $\frac ab=1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\frac18+\frac19-\frac1{10}$, such that $a,b$ have no common divisor greater than 1, find $a$.

In the following figure, $ABCD$ is a square, $BD$ is parallel to $EC$ and $BD=BE$. Find $\angle BEC$ in degrees.

Motorcyclist A travels from Town P to Town Q whereas motorcyclist B travels from Town Q to Town P. Both motorcyclists started their journey at the same time. They met at a location which was 80 km from Town Q. Both motorcyclists continued their journey in their respective directions after they met. When both motorcyclists reached their respective destinations, they immediately travelled back to their respective starting points. If both motorcyclists met again, without overtaking, at a location which was 60 km from Town P, find the distance between Town P and Town Q.

2017 students arrange themselves in a single file. In the first round of counting, they number themselves $1,2,3,1,2,3,1,2,\ldots$ from left to right. In the second round, they number themselves $1,2,3,4,5,1,2,3,4,5,1,2,\ldots$ from right to left. Find the number of students whose difference between their numbers in the first and second rounds is 1.

The figure below is made up of three squares of sides 12 cm, 18 cm and 24 cm respectively. Find the area of the shaded triangle.

A pump is filling a tank with water at a constant rate while water in the tank is also drained by 3 identical pipes. Starting from a full tank with water being pumped into it, the tank can be completely drained by the 3 identical pipes in 35 minutes. If an additional identical pipe is installed, a full tank can be completely drained in 20 minutes while water is pumped into it. How many minutes will it take for the pump to fill an empty tank to its brim without any draining pipe?

How many fractions of the form $\frac ab$ are there, where $a,b$ have no common factors larger than 1, such that $b=a+6$ and $\frac ab<\frac{2017}{2023}$?

There are 6 red points labelled $R_1,R_2,R_3,R_4,R_5,R_6$ and 6 green points labelled $G_1,G_2,G_3,G_4,G_5,G_6$. Using straight lines, each red point is connected to at least one green point and each green point is connected to at least one red point. The numbers of green points connected to $R_1,R_2,R_3,R_4,R_5$ are 5,4,3,2,2 respectively. The numbers of red points connected to $G_1,G_2,G_3,G_4,G_5$ are 4,3,2,1,1 respectively. Find the number of red points connected to $G_6$.