A point $D$ lies on the side $AC$ of a triangle $ABC$. Triangle $ADB$ is isosceles with $DA=DB$. Triangle $DBC$ is also isosceles with $BC=BD$. All angles in triangles $ADB$ and $DBC$ are an integer number of degrees. What is the difference between the largest and smallest values that angle $ADB$ could have?
Numerical answer only; a non-negative integer less than 1000.
Amy has 14 cousins aged $2,3,4,\ldots,15$ respectively. She also has some cards separately numbered $16,17,18,\ldots,k$, for some integer $k$. Amy manages to give one card to each cousin so that the number on the card is a multiple of that cousin’s age. Find the least $k$ for which this is possible.
Numerical answer only; a non-negative integer less than 1000.
There are 5 lily pads on a pond, arranged in a circle. A frog can only jump from each lily pad to an adjacent lily pad on either side. How many ways are there for the frog to start on one of these lily pads, make 11 jumps, and end up where it started?
Numerical answer only; a non-negative integer less than 1000.
Find the sum of all (not necessarily distinct) values of $a$ over all triples $(a,b,c)$ of real numbers that satisfy the equations:
$$(a+b)(c+1)=22\quad(1),$$
$$(a+c)(b+1)=22\quad(2),$$
$$(b+c)(a+1)=22\quad(3).$$
Numerical answer only; a non-negative integer less than 1000.
A triangle has sides of length $x,y,20$ where $x\gt y\gt 20$ and $x,y$ are integers. Let $h$ be the length of the altitude of the triangle from the side of length 20. Let $h_x,h_y$ be the lengths of the altitudes from the sides of length $x,y$ respectively. These altitudes satisfy $h=h_x+h_y$. Find the perimeter of this triangle.
Numerical answer only; a non-negative integer less than 1000.
A word is a sequence of zero or more letters taken from the set $\{A,B,C,D,E,F,G,H,I\}$. Two words are said to be related if one can be obtained from the other by a sequence of the following operations:
(a) swapping two adjacent letters,
(b) deleting two adjacent letters which are the same,
(c) inserting two adjacent letters which are the same.
Thus, for instance, BACA and BCAA are related by swapping adjacent letters A and C, and BCAA is related to BC by deleting or inserting two adjacent letter As. Note that the empty word with zero letters is related to any word consisting of just two adjacent letters.
What is the maximum number of words in a set in which no word is related to any other?
Numerical answer only; a non-negative integer less than 1000.
Identical wind turbines are equally spaced in a straight line on level ground. Each turbine tower is a vertical cylinder of radius 1 metre. Let $\mathcal L$ be the line through the base centres of the towers. There is an observer fixed at $O$ on the same level ground as the towers. Let $A$ be the point on $\mathcal L$ that is closest to $O$.
If $A$ is the base centre of a tower and $OA=15$ metres, what is the maximum number of towers that are completely visible to the observer over all possible distances $d$ metres between the base centres of adjacent towers? [5 marks]
Investigation: If $OA=16$ metres and $A$ is not the base centre of a tower, what is the maximum number of towers that are completely visible to the observer? [4 bonus marks]
Written reasoning is required. The reference answer is not an automatic assessment of a student proof.