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) = 5x

[IrMO 2000, Paper 1, Question 3]

^{13}+ 13x^{5}+ 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

y

^{2}= (x + 8)(x

^{2}+ 2);

y

^{2}= (8 + 4x)y + 5x

^{2}- 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?

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?

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