2

^{n}= a! + b! + c!

in positive integers a, b, c and n.

(Here, ! means "factorial".)

[IrMO 2001 Paper 1 Question 1]

Find, with proof, all solutions of the equation

2^{n} = a! + b! + c!

in positive integers a, b, c and n.

(Here, ! means "factorial".)

[IrMO 2001 Paper 1 Question 1]

2

in positive integers a, b, c and n.

(Here, ! means "factorial".)

[IrMO 2001 Paper 1 Question 1]

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?

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

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.

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?

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]

Consider all parabolas of the form y = x

[IrMO 2000, Paper 1, Question 5]

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]

(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]

Let f(x) = 5x^{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]

[IrMO 2000, Paper 1, Question 3]

Solve the system of (simultaneous) equations

y

y

[IrMO 1999, Paper 2, Question 1]

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?

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?

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]

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.

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]

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?

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]

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.

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.

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).

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).

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?

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?

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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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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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)?

**Feel free to comment, ask questions and even check your answer in the comments box below powered by Google+.**

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

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If you enjoy using this website then please consider making a donation - every little helps :-)

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