Working Hours: Upper Primary Mathematics Competition Question

Mr Workalot wakes up at 5:45 am. He sets off to work and arrives at (x + 1) am, he then leaves work at (x – 3) pm.

How many hours does Mr Workalot work today?

Money Sharing: Lower Secondary Mathematics Competition Question

Three friends are sitting at a table and wish to play a money-sharing game. Alice gives half her money to Brenda; then Brenda gives half her new total to Carla; and finally, Carla gives half her current total to Alice.

If the three friends each end up with $10, how much money did they each have at the start?

IrMO 1988 P2 Q3: More Bus Stops! : Upper Secondary Mathematics Competition Question

The question I posted for PMQ30 is a simplified version of a real maths competition problem. This is from Paper 2 of the IrMO 1988; such papers have only three questions so are closer to the IMO than Paper 1. Having said that, I think most solvers will need to go through the process of solving PMQ30 to understand how to construct more complicated networks.

The Question

A city has a system of bus routes laid out in such a way that

(a) there are exactly 11 bus stops on each route;

(b) it is possible to travel between any two bus stops without changing routes;

(c) any two distinct bus routes have exactly one bus stop in common.

What is the number of bus routes in the city?

[IrMO 1988 Paper 2 Question 3]

Prize Maths Quiz: A Bus Network Puzzle (PMQ30)

An airport has a system of bus routes to shuttle passengers between terminals. The system is laid out in such a way that:

(a) there are exactly 3 bus stops on each route;

(b) it is possible to travel between any two bus stops without changing routes;

(c) any two distinct bus routes have exactly one bus stop in common.

What is the number of bus routes in the airport?

The diagram shows the fairly trivial case in which each route has just 2 bus stops; in this case, we only need 3 routes to satisfy all the conditions.

Now, there is an extension to this question. In the past, I have occasionally had a choice of two questions for a PMQ. With hindsight, I suspect this has led to some confusion, so I’m posting the extension here.

Extended Domino Triangles: Upper Secondary Mathematics Competition Question

This is an extension to the previous domino triangles problem; I hope Chris Breederveld doesn’t mind me messing with his puzzle!

In this case, take a full set of dominoes but don’t remove the tiles with blanks. The tiles with blanks can still be used to denote a zero. For example, the tile [2/0] can be used as the fraction 0/2 but obviously not as 2/0! This means that the tile [0/0] is the only one that must be removed as it is effectively useless.

Now the question is essentially the same as before. Starting with the tile [6/6] at the top of the triangle, every tile-fraction is the sum of the two tile-fractions below it. How many unique solutions can you find? Treat reflections as one distinct solution.

The diagram below shows the start of one possible solution. Compare this with the diagram in the previous question.