Show that a disc of radius 2 can be covered by seven (possibly overlapping) discs of radius 1.

[IrMO 1998 Paper 1 Question 4]

Show that a disc of radius 2 can be covered by seven (possibly overlapping) discs of radius 1.

[IrMO 1998 Paper 1 Question 4]

P is a point inside an equilateral triangle such that the distances from P to the three vertices are 3, 4 and 5, respectively. Find the area of the triangle.

[IrMO 1998 Paper 1 Question 2]

[IrMO 1998 Paper 1 Question 2]

The diagram shows the numbers 1 to 25 inserted into a table. Draw a circle round some of the prime numbers in such a way that every row and every column has only one circled prime.

a) What is the sum of your circled primes?

b) Prove that your solution is unique.

An integer n is called 'paladin' if the number of positive divisors of n is equal to the number of digits of n.

a) Determine all three-digit 'paladin' numbers.

b) Determine a 'paladin' number with four digits that is less than 1300.

[OPM 2012 Level Beta Paper1 Q2][Original in Portuguese. My translation into English.]

[OPM = Olimpiada Paulista de Matematica, Brazil]

The same question is repeated in the Gamma Level paper (borderline middle/upper secondary) but with a harder part (b).

b2) Determine a 'paladin' number with six digits that is less than 110000.

[OPM 2012 Level Gamma Paper 1 Q1]

Alice’s weekend homework was on number bases. It was all pretty simple stuff and didn’t take any hard thinking at all. So Alice started to play around with the numbers to see if anything interesting happened.

The number 8 in base 10 could be written as 1000 base 2, 22 base 3, 20 base 4, 13 base 5, and so on until it gets stuck at 8 in base 9, 10, 11, 12 and any other base. Was there a base 1? Alice thought about this as the school book was silent on this matter. If there was a base 1 then 8 would be written as 11111111. Imagine writing 88 in base 1! Not only would it be really long and boring to do but it would also be long and boring to check that it really did have 88 digits, and not 87 or 89. It dawned on Alice why we use base 10 for counting. However, it didn’t make sense why clocks go up to 12 rather than 10, but she was happy to have had one bright idea this morning and didn’t want to spoil that smiley feeling.

Alice went back to doodling numbers and found something else: 13 base 4 was equal to 7 (in base 10), 22 base 4 = 10, 31 base 4 = 13, 103 base 4 = 19, 112 base 4 = 22. Patterns. Alice loved patterns, but this one seemed to have gaps in it. This needed a plan and lots of calculations. The idea that Alice was trying to formulate in her mind was the following.

Let’s take a whole number B and express it as A base N in such a way that N is the sum of the digits of A. If A base N = B base 10, then B is a ‘basiq’ number. If B cannot be expressed in this way then it is a non-basiq number. How many non-basiq numbers are there between 1 and 100 inclusive?

We have just seen that 13 base 4 = 7 and that 1+3=4 so that 7 is a basiq number. However, there is no way to write 8 in the same form so 8 is non-basiq. Alice quickly discovered a simple rule that cut down her calculations... and then another! She had completed her puzzle too quickly! There was, however, a lingering question: was there a highest non-basiq number or did they carry on getting ever-bigger? This sounded like a challenge for Professor Pailyn.

Can you help Alice before she visits the Professor?

But firstly, can you solve Alice’s puzzle, to find all non-basiq numbers between 1 and 100 inclusive?

Show that no integer of the form xyxy in base 10, where x and y are digits, can be the cube of an integer.

Find the smallest base b > 1 for which there is a perfect cube of the form xyxy in base b.

[Irish MO (IrMO) 1998 Paper 1 Question 3]

Find the smallest base b > 1 for which there is a perfect cube of the form xyxy in base b.

[Irish MO (IrMO) 1998 Paper 1 Question 3]

Let S be the set of all natural numbers n satisfying the following conditions:

(i) n has 1000 digits;

(ii) all the digits of n are odd, and

(iii) the absolute value of the difference between adjacent digits of n is 2.

Determine the number of distinct elements in S.

[Irish MO (IrMO) 1997 Paper 2 Q4]

A speed chess tournament is organised according to the following rules. Each round consists of all one-game matches played between two players. A player is eliminated from the tournament once that player has lost 2 games. No games are drawn. The draw for each round is done by randomly drawing names from those players still in the tournament. If there are an odd number of players in a round, a bye* is given to one player randomly drawn from the set of players with the smallest number of losses and the least number of byes already received; this takes place before the draw for that round.

Given that 33 players start the tournament, what is the minimum number of games needed to guarantee reaching one outright winner?

*Note that a bye means the player does not play in that round but still passes onto the next round.

Given that 33 players start the tournament, what is the minimum number of games needed to guarantee reaching one outright winner?

*Note that a bye means the player does not play in that round but still passes onto the next round.

The product 11x11=121 is true in every number base greater than 2.

What is the largest 3-digit number that can be expressed as the product of two 2-digit numbers in such a way that the whole calculation is true in every number base greater than 5.

Write out the full calculation.

What is the largest 3-digit number that can be expressed as the product of two 2-digit numbers in such a way that the whole calculation is true in every number base greater than 5.

Write out the full calculation.

Alice had a fever. She was in bed, her head felt like a cushion and she was tired of being tired. She tried to read but soon drifted off into sleep. But it was not a restful sleep; her mind had wandered into a landscape of numbers.

She saw trains of numbers racing round an imaginary roller-coaster. She saw snakes of numbers biting their tails and rolling away like monocycles on a mission. The whole world seemed like a sea of numbers, some grew bright and shiny as if demanding attention while others faded away into the background. If they were trying to show her some deep and meaningful patterns they never kept still long enough for Alice to figure them out. She somehow felt like they were mocking her like spritely flighty things that were always out of reach.

Consider an isosceles triangle with sides (m, m, 2m-2). If the area of the triangle is 360, find the value of m.

A large solid 3x3x3 cube is made from white unit cubes. The six faces of the large cube are then painted blue. After waiting for the paint to dry, the unit cubes are rearranged so that the faces of the large cube are all white again. All the faces of this large cube are then painted red.

Now, how many unit cube faces are still white?

Alice was waiting for her friend Brenda to arrive; she was late. Alice had prepared some games – Professor Pailyn seemed to have so many games, yet no children to actually

“Professor, look, I can make patterns like this and I can do it with just three colours. Isn’t there some idea that you need four colours?”

“Yes, I think you mean the four colour map theorem.” He eased himself into a comfy armchair.”Take any plane, such as a sheet of paper, draw some regions – you can try to make them as complicated as you please. Then, if you try to colour each region in such a way that no two adjacent regions have the same colour, you find that only a maximum of four colours are needed.”

The lengths of the sides of an isosceles triangle are all whole numbers and the perimeter is 16 units.

Find the largest possible area of such a triangle.

(You may leave your answer in an exact form.)

Find the largest possible area of such a triangle.

(You may leave your answer in an exact form.)

A scalene triangle has a perimeter of 19 units. If all the side lengths are odd numbers, what is the product of the three lengths?

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

[APMO 1992 Q5]

[APMO 1992 Q5]

Determine all pairs (h, s) of positive integers with the following property:

If one draws h horizontal lines and another s lines which satisfy

(i) they are not horizontal,

(ii) no two of them are parallel,

(iii) no three of the h + s lines are concurrent,

then the number of regions formed by these h + s lines is 1992.

[APMO 1992 Q4]

Let n be an integer such that n > 3. Suppose that we choose three numbers from the set {1, 2, ..., n}. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations.

(a) Show that if we choose all three numbers greater than n=2, then the values of these combinations are all distinct.

(b) Let p be a prime number such that p ≤ √n. Show that the number of ways of choosing three numbers so that the smallest one is p and the values of the combinations are not all distinct is precisely the number of positive divisors of (p - 1).

[APMO 1992 Q3]

“Hello. I’m Alice. I live next door. I brought you some cake as a... erm... hello present!” She beamed to hide her tongue-tied introduction.

“Well, hello Alice. My name is Professor Pailyn.”

“Professor Pie-Lynne, pleased to meet you.” Alice felt like a little pixie shaking hands with a large bear, but he had a broad smile under that grizzly beard and looked friendly enough.

“So... what work do you do, Professor?” Alice was trying to be nosey to see what the inside of the house looked like, but she couldn’t see round the Professor - nor through him, of course.

“I’m a mathematician.”

A triangle with sides a, b and c is given. Denote by s the semiperimeter, that is, s = (a+b+c)/2. Construct a triangle with sides s-a, s-b, and s-c. This process is repeated until a triangle can no longer be constructed with the side lengths given.

For which original triangles can this process be repeated indefinitely?

[APMO 1992 Q1]

For which original triangles can this process be repeated indefinitely?

[APMO 1992 Q1]

The product below uses 7 different digits, excluding a zero. Replace each symbol with the right digit to make the product correct. How many solutions are there?

Now, if we did allow a zero, are there any more solutions?

Have fun!

Now, if we did allow a zero, are there any more solutions?

Have fun!

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