## 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]

## 27 Jun 2014

### IrMO 2002 P2 Q2: Upper Secondary Mathematics Competition Question

Suppose n is a product of four distinct primes a, b, c, d such that

(a) a + c = d;

(b) a(a + b + c + d) = c(d - b);

(c) 1 + bc + d = bd.

Determine n.

[IrMO 2002, Paper 2, Question 2]

## 22 Jun 2014

### IrMO 2000 P1 Q3: Upper Secondary Mathematics Competition Question

Let f(x) = 5x13 + 13x5 + 9kx. Find the least positive integer k such that 65 divides f(x) for every integer x.

[IrMO 2000, Paper 1, Question 3]

## 20 Jun 2014

### IrMO 1999 P2 Q1: Upper Secondary Mathematics Competition Question

Solve the system of (simultaneous) equations

y2 = (x + 8)(x2 + 2);

y2 = (8 + 4x)y + 5x2 - 16x - 16:

[IrMO 1999, Paper 2, Question 1]

### Domino Products: Lower Secondary Mathematics Competition Question

Jason is given a set of domino tiles. He is asked to remove all the tiles with a blank square, leaving him with 21 distinct dominoes. He is asked to place two tiles within a 2x2 square grid in such a way that the products of the two numbers along the diagonals are equal.

The diagram shows one solution. In this case, we have 1x4 = 2x2 and we have used the tiles [1,2] and [2,4]. Any solution that is a rearrangement of the same pair of tiles is ignored - we are just interested in which tiles are used.

How many distinct pairs of tiles will solve the above problem?

## 14 Jun 2014

### Primes in a Triangle: Lower Secondary Mathematics Competition Question

Place nine distinct prime numbers into the grid in such a way that the sums of the four numbers along each side are all equal to each other.

What is the smallest possible such sum?

### IrMO 1999 P2 Q2: Upper Secondary Mathematics Competition Question

A function f : N --> N (where N denotes the set of positive integers) satisfies

(a) f(ab) = f(a)f(b) whenever the greatest common divisor of a and b is 1,

(b) f(p + q) = f(p) + f(q) for all prime numbers p and q.

Prove that f(2) = 2, f(3) = 3 and f(1999) = 1999.

As we're in the year 2014, calculate f(2014).

[adapted from IrMO 1999, Paper 2, Question 2]

## 7 Jun 2014

### Number of Divisors: Middle Secondary Mathematics Question

Let d(n) be the number of positive divisors of an integer n. For example, d(15) = 4.

Find the smallest positive value of n such that d(n) = d(n+1) = 6.

### IrMO 1999, P2 Q4: Upper Secondary Mathematics Competition Question

Find all positive integers m with the property that the fourth power of the number of (positive) divisors of m equals m.

[IrMO 1999, Paper 2, Question 4]

## 1 Jun 2014

### Three Dice: Upper Primary Mathematics Competition Question

Trinity rolled three fair six-sided dice, each numbered 1 to 6.

If the sum of her three numbers is 14, what is the highest possible product she could get?

## 31 May 2014

### IrMO 1999 P1 Q5: Upper Secondary Mathematics Competition Question

Three real numbers a, b, c with a < b < c, are said to be in arithmetic progression if c - b = b - a.

Define a sequence u(n), n = 0, 1, 2, 3, ... as follows: u(0) = 0, u(1) = 1 and, for each n > 0, u(n+1) is the smallest positive integer such that u(n+1) > u(n) and {u(0), u(1),... u(n), u(n+1)} contains no three elements that are in arithmetic progression.

Find u(100).

[Irish MO, Paper 1, Question 5, 1999]

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

Let N be a natural number with the property that it is the sum of 3 consecutive natural numbers, of 4 consecutive naturals and also of 5 consecutive naturals.

Find the smallest value of N such that the 3 sequences above are disjoint, that is, there is no number that is in more than one sequence.

## 25 May 2014

### Pairs of Unit Fractions: Upper Secondary Mathematics Competition Question

For each positive integer n, let S(n) be the set of ordered pairs (x,y) of positive integers such that

1/n = 1/x + 1/y

and let T(n) be the number of ordered pairs in S(n).

For example, for n=2, S(2)={(3,6), (4,4), (6,3)} and hence T(2)=3.

a) Determine T(n) for all n.

b) Hence, calculate T(2014).

## 24 May 2014

### Four Perimeters: Upper Primary Mathematics Competition Question

A rectangle is divided into four smaller rectangles using two perpendicular lines, as shown in the diagram.

The number inside each small rectangle indicates the length of its perimeter. The diagram is not drawn to scale.

What value should go into the empty rectangle?

### Six Eggs in a Box: Lower Secondary Mathematics Competition Question

Edward likes eggs. Not necessarily to eat them; he likes playing with them too. Today, he has taken eggs from different boxes so that he has 3 white eggs and 3 brown eggs sitting in a 6-egg box. He is thinking about how many patterns he can make with his 6 eggs.

How many different arrangements can Edward make using all 6 eggs in the one box?

As the box has a lid, any similar patterns under rotation count as two distinct arrangements.

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### Divisibility by 99: Middle Secondary Mathematics Competition Question

You have nine cards numbered from 1 to 9. If you pick each card randomly and lay them out in order, find the probability that the resulting 9-digit number is divisible by 99.

Express this probability as m/n, where m and n are relatively prime. What is the sum (m + n)?

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