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?

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?

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?

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?

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]

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).

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)?

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

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A Tangled Peg-Board: Professor Pailyn's Mathematical Quest PMQ46

Alice was playing around with a peg-board and a jar of elastic bands. She was trying to think up a devilish problem to give Brenda when she finally turned up. Alice had made a real tangled mess and was annoyed with herself for then having to take all the rubber bands off... and then one peg snapped off! She stared at the board, as if willing the peg to jump right back into its rightful place; but it didn't, of course.

However, this now meant that Alice could play with a smaller board. She laid out a perimeter so that she was left with a board of 6 by 6 pins (as shown in the diagram). She had thought of how many elastic bands she'd need if she joined together every pair of pins that were a whole number distance apart, but that had ended up badly. So then she had another idea: join together every pair of pins that are a prime number distance apart. That sounded better! Assuming, of course, that the distance between adjacent pins was a unit length.

So, how many elastic bands will Alice need this time? Do you think she'll have enough of them?!

Houses in a Row: Professor Pailyn's Mathematics Quest PMQ44

“Look, I’ve invented a problem that I can’t figure out!” Alice sounded exasperated. “I thought it was going to be easy, but it isn’t. Imagine four houses placed on a map and each connected to all the others with a straight road. Now, I was wondering whether the length of every road could be a whole number. That bit is easy! So then I thought about whether the length of every road could be a different whole number. That’s when I got stuck!” Alice slumped in her chair, just to illustrate her defeat.

“Your problem does, indeed, have solutions.” Soothed Professor Pailyn, “But the mathematics needed to find them may take a while to go through. It just means learning some new topics that you probably haven’t yet done in school. We can do it, but your problem has made me think of a similar situation.” The Professor took some paper and sat down next to Alice.

OEMO 2001 Advanced Q4: Upper Secondary Mathematics Competition Question

Let A₀ = {1, 2} and for n > 0 let A_n be the set of all numbers that are either elements of A_n-1 or can be represented as the sum of two distinct elements of A_n-1.

Further let a_n = |A_n| be the number of elements of A_n.

Determine a_n as a function of n.

[OEMO 2001 Advanced Q4]

A Necklace Puzzle: Professor Pailyn's PMQ43

Alice’s mother had bought her a set of beads with which to make some necklaces. Alice had originally frowned upon this as an attempt to give her something girlie-ish. However, she did enjoy making patterns so started playing with the beads. But on this particular day, Alice was getting bored with her pattern-making skills.

"I need to find a really clever pattern; something to impress Zeta. I know just the person to ask."

-=*=-

“OK, let’s start with a simple example. Pick just two colours...”

“Pink and... orange!” Alice interjected.

“Isn’t that a touch... garish?” enquired the Professor.

“It’s not garish... it’s colourful!” beamed Alice. It was, indeed, going to be hideous but it was for her cousin.

“Alright, then pick a maximum length for the beads – let’s pick 3 to make the numbers small.”

“Three! That’s not a necklace, that’s an earring!” Alice was having fun mocking the good Professor. He knew this and happily played along.

“Wait a minute, my impatient child! Now, we’re going to make a necklace where all possible 3-bead permutations exist. How many beads do you think you’ll need?” It was the Professor’s turn to smile with his ‘show-me-how-clever-you-are’ question.

IrMO 1997 P2 Q4: Upper Secondary Mathematics Competition Question

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 Chess Tournament: Middle Secondary Mathematics Competition Question

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.

APMO 1992 Q4: Upper Secondary Mathematics Competition Question

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]

APMO 1992 Q3: Combinatorics: Upper Secondary Mathematics Competition Question

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]

APMO 1989 Q4: Combinatorics: Upper Secondary Mathematics Competition Question

Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least



triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.


[APMO 1989 Q 4]

JBMO 2013 Q4: A Game of Sums: Middle Secondary Mathematics Competition Question

Let n be a positive integer. Two players, Alice and Bob, are playing the following game:

• Alice chooses n real numbers, not necessarily distinct;

• Alice writes all pairwise sums on a sheet of paper and gives it to Bob (there are n(n-1)/2 such sums, not necessarily distinct);

• Bob wins if he finds correctly the initial n numbers chosen by Alice with only one guess.

Can Bob be sure to win for the following cases?

a) n = 5 b) n = 6 c) n = 8

Justify your answer(s).

[For example, when n = 4, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

[Junior Balkan MO 2013 Problem 4][Note that the original paper has 4 problems to be done in 4.5 hours]

BdMO 2012: Tajingdong mountain problem: Primary Mathematics Competition Question

When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining 11 times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent 16 mornings and 13 afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?


[Bangladesh BdMO 2012. This question appeared in the Primary, Junior and Secondary competitions.]

APMO 1991 Q2: Upper Secondary Mathematics Competition Question

Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane.

Can you find a special case with exactly 1991 red points?


[APMO 1991 Q2]