The diagram below shows nine circles joined by seven lines, with three circles on each line. Replace each of $a,b,c,d,e,f,g,h$ and $k$ by a different one of $1,2,3,4,5,6,7,8$ and $9$, such that the number in the middle circle of a solid line is equal to the sum of the numbers in the other two circles, and the number in the middle circle of a dotted line is equal to the difference of the numbers in the other two circles.
Original answer template:
Submission guidance: Supply the completed diagram only. The source does not require a written proof for this odd-numbered question. This graphical response is manually graded, with all-or-nothing credit.
When the digits of a three-digit number $x$ are written in reverse order, we obtain a number $y$ such that $x+2y=2016$. Determine the sum of all possible values of $x$.
Submission guidance: Give a full solution, as required for this even-numbered problem by the examination cover.
How many of the first 2016 positive integers can be expressed in the form $1+2+\cdots+(k-1)+mk$, where $k$ and $m$ are positive integer? For example, we have $6=1+2+3\times1$ and $11=1+2\times5$.
A circle with diameter $AB$ intersects a circle with centre $A$ at $C$ and $D$. $E$ is the point of intersection of $AB$ and $CD$. $P$ is a point on the second circle such that $PC=16$ cm, $PD=28$ cm and $PE=14$ cm. Find the length, in cm, of $PB$.
Submission guidance: Give a full solution, as required for this even-numbered problem by the examination cover.
The diagram below shows an arrangement of 20 numbered circles. Note that circles 3, 9, 12 and 18 determine a square. What is the minimum number of circles we have to remove so that no four remaining circles determine a square?
A Mathematics test consists of 3 problems, each worth an integral number of marks between 1 and 10 inclusive. Each student scores more than 15 marks, and for any two students, they obtain different numbers of marks for at least one problem. Find the maximum number of students.
Source clarification: The organizer's Traditional Chinese edition explicitly says that the score obtained on each problem is an integer from 1 to 10. This positive-integer score domain is used here; the English wording above is retained.
Submission guidance: Give a full solution, as required for this even-numbered problem by the examination cover.
Let $x,y$ and $z$ be positive real numbers such that
$$\sqrt{16-x^2}+\sqrt{25-y^2}+\sqrt{36-z^2}=12.$$
If the sum of $x,y$ and $z$ is 9, find their product.
What is the largest number of integers that may be selected from 1 to 2016 inclusive such that the least common multiple of any number of integers selected is also selected?
Submission guidance: Give a full solution, as required for this even-numbered problem by the examination cover.
Dissect the six figures in the diagram below into twelve pieces, each consisting of five squares, such that no two of the twelve pieces are identical up to rotation and reflection.
Original answer template:
Submission guidance: Supply the six completed dissection diagrams only; no written proof is required by the source for this odd-numbered problem. Each piece must be connected along square edges. The black cell is excluded. Any dissection satisfying all conditions is accepted by manual, all-or-nothing grading.
