## 13 Jul 2014

### IrMO 2001 P1 Q1: Upper Secondary Mathematics Competition Question

Find, with proof, all solutions of the equation

2n = a! + b! + c!

in positive integers a, b, c and n.

(Here, ! means "factorial".)

[IrMO 2001 Paper 1 Question 1]

### Matching Octagons: Middle Secondary Mathematics Competition Question

The image shows two regular octagons, each has 4 red and 4 blue beads, one placed on each vertex.

We say that there is a 'match' if a vertex has the same colour in both octagons. In the diagram, we can see that the upper-right vertices are both blue and the lower-right vertices are both red - all the other vertex pairs have different colours. In this case, we have a matching value of 2.

However, if we rotate the inner octagon by 45 degrees clockwise then all the vertices match, giving us a maximum matching score of 8 for these two arrangements.

Now, if we randomly allocate the beads to the two octagons (each with 4 of each colour) and we are allowed to rotate one of the octagons, what is the smallest guaranteed maximum matching score?

### Sums of Factorials: Middle Secondary Mathematics Competition Question

Let N = a! + b!

Find all solutions (a,b) such that N is divisible by 11 and both a and b are positive integers less than 11, with a ≤ b.

How many solutions are there?

You may leave your answers in the form n!, where n! = n.(n-1).(n-2)... 3.2.1.

## 5 Jul 2014

### Beads on a Hexagon: Lower Secondary Mathematics Competition Question

Six beads are arranged at the corners of a regular hexagon; 3 are orange and 3 green. All arrangements that are rotational symmetries of each other count as one unique arrangement.

Using all six beads, how many unique arrangements are there?

### IrMO 2000 P2 Q4: Upper Secondary Mathematics Competition Question

Prove that in each set of ten consecutive integers there is one which is coprime with each of the other integers.

For example, taking 114, 115, 116, 117, 118, 119, 120, 121, 122, 123 the numbers 119 and 121 are each coprime with all the others. [Two integers a, b are coprime if their greatest common divisor is one.]

[IrMO 2000 Paper 2 Question 4]

## 4 Jul 2014

### IrMO 2000 P1 Q5: Upper Secondary Mathematics Competition Question

Consider all parabolas of the form y = x2 + 2px + q (p, q real) which intersect the x- and y-axes in three distinct points. For such a pair p, q let C(p,q) be the circle through the points of intersection of the parabola y = x2 +2px+q with the axes. Prove that all the circles C(p,q) have a point in common.

[IrMO 2000, Paper 1, Question 5]