IMSO 2014 — Math Division — Exploration

Each of the $36$ cells in a $6\times6$ table is to be filled with numbers $0,1$ or $2$. The sum of the six numbers in each row and each column is then computed. How do you fill the cells to get the maximum number of different values among the $12$ sums?

In the Indonesian village Sevenia, each bill is worth a number of rupiahs equal to a power of $7$, namely, $7^0=1$, $7^2=49$, $7^3=343$, $7^4=2401$, $7^5=16807$, $7^6=117649$, $7^7=823543$ and so on. A millionaire has some bills worth a total of exactly $1000000$ rupiahs, but at most $8$ bills of any kind. No two millionaires have exactly the same combination of bills. What is the maximum number of millionaires in Sevenia?

The diagram below shows two figures connected and constructed by using six units of equilateral triangles satisfying the following two conditions:

(1) When two equilateral triangles are joined together, an entire edge of one must match up with an entire edge of the other.

(2) The final figure formed is in one connected piece.

Draw all other such diagrams.

Write down all positive integers $n\geq9$ which are divisible by every positive integer not greater than $\frac n9$.

A magic triangle consists of several circles in a triangular configuration and are filled with consecutive integers starting from $0$. The integer written in a circle that is supported by two other circles below it is the remainder when the sum of the integers in the supporting circles is divided by the number of circles of the triangle. Moreover, the integer in the bottom leftmost circle should be always less than the integer in the bottom rightmost circle. The diagram below shows an example of a triangle with three circles such that the only division $(1+2)\div3$ leaves a remainder of $0$.



(a) Write down all magic triangles with six circles. (2 marks)

(b) Write down all magic triangles with ten circles. (4 marks)

The three shapes in the diagram below are called F, I and W respectively.



(a) Use two copies of F to make a diagram, and then two copies of W to make the same diagram. No overlap is allowed. (1 mark)

(b) Use several copies of F to make a diagram, and then the same number of copies of I to make the same diagram. No overlap is allowed. (2 marks)

(c) Use several copies of W to make a diagram, and then the same number of copies of I to make the same diagram. No overlap is allowed. (3 marks)