27 Nov 2013

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?!

21 Nov 2013

MEMO 2012 Q2: Upper Secondary Mathematics Competition Question

Let N be a positive integer. A set S is a subset of {1, 2, ..., N} and is called 'allowed' if it does not contain three distinct elements a, b, c such that a divides b and b divides c.

Determine the largest possible number of elements in an allowed set S.

[MEMO 2012 Problem I-2]
[MEMO = Middle European Mathematical Olympiad]

19 Nov 2013

15 Nov 2013

Integral Network: Professor Pailyn's Mathematical Quest PMQ45

“That was a bit complicated!” Exclaimed Alice, “But what about my original problem? Can we do that now?”

“I think it’s time to recharge those brain cells... with some tea and fruit cake.” Professor Pailyn walked straight past Alice, carrying the tea-tray into the garden.

Alice enjoyed elevenses; it was almost better than breakfast. Alice had never had a proper stroll around the Professor’s garden. As she walked through it with her eyes, she noticed some peculiar structures: a cube within a cube standing on one corner, and what appeared to be a leafy halo suspended in mid-air betwixt two egg-shapes. She wanted to go and find the hidden wires but was too busy eating cake.

“So... let’s have a little think about your problem. You have four points on a plane, all connected to each other by straight lines. No three points lie on the same straight line so that we have six distinct line segments. You want all six lines to be different whole numbers. Is that right?”

14 Nov 2013

Sliced Parallelogram: OEMO 2003 Advanced Q4: Upper Secondary Mathematics Competition Question

In a parallelogram ABCD, let E be the midpoint of the side AB and F the midpoint of BC. Let P be the intersection point of the lines EC and F D.

Show that the segments AP, BP, CP and DP divide the parallelogram into four triangles with areas in 1 : 2 : 3 : 4 ratio.

[OEMO 2003 Federal Advanced Q4]

9 Nov 2013

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 Q1: Upper Secondary Mathematics Competition Question

Let n be an integer and S(n) be the sum of the 2001 powers of n with exponents 0 through 2000.
That is,

Determine the final digit (i.e., the ones-digit) in the decimal expansion of S(n).

[OEMO 2001 Advanced Q1]

8 Nov 2013

OEMO 2001 Advanced Q4: Upper Secondary Mathematics Competition Question

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

Further let an = |An| be the number of elements of An.

Determine an as a function of n.

[OEMO 2001 Advanced Q4]

7 Nov 2013

Peculiar Pentagon: Upper Secondary Mathematics Competition Question

In a convex pentagon ABCDE the areas of the triangles ABC, ABD, ACD and ADE are all equal to the same value F.

What is the area of the triangle BCE?

[OEMO 2001 Advanced Q3]

Between Circles: Middle Secondary Mathematics Competition Question

Three circles of radius 6 cm are arranged as shown in the diagram. A band is placed tightly around the circles. This creates four regions that are bounded by the band but that are external to the circles - these are shown in dark blue.

Find the area highlighted in dark blue.

You may leave your answer as an expression in π.

Ratio of Squares: Lower Secondary Mathematics Competition Question

A large square has a diagonal drawn across it, thereby creating two isosceles triangles. Within each triangle is drawn a smaller square as shown in the diagram.

What is the ratio of the areas of square A to square B?

1 Nov 2013

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.

30 Oct 2013

Circled Primes: Upper Primary Mathematics Competition Question

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.

OPM 2012 Beta P1 Q2: Middle Secondary Mathematics Competition Question

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]

26 Oct 2013

Basiq Numbers: PMQ42

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?

23 Oct 2013

IrMO 1998 P1 Q3: Cubes and Bases: Upper Secondary Mathematics Competition Question

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]

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.

19 Oct 2013

Pailyn's Palindromic Pairs: PMQ41

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.

16 Oct 2013

Area of Isosceles Triangle: Middle Secondary Mathematics Competition Question

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

Colouring Cubes: Lower Secondary Mathematics Competition Question

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?

12 Oct 2013

Circle Patterns With Four Colours: Professor Pailyn's Mathematical Quiz PMQ40

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 play them. To pass the time, Alice was making patterns from a collection of circular plastic counters. Professor Pailyn walked in with a jug of what looked like wine but was most probably apple juice. Before Alice could make a sound, he walked out again. He quickly returned with some glasses.

“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.”

8 Oct 2013

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]

7 Oct 2013

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]

4 Oct 2013

A Borromean Number Puzzle: Professor Pailyn's Mathematical Quiz PMQ39

Borromean rings in detail of coat of arms of the House of Borromeo
“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.”

3 Oct 2013

APMO 1992 Q1: A Semiperimeter Problem: Upper Secondary Mathematics Competition Question

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]

2 Oct 2013

A Product Number Puzzle: Lower Secondary Mathematics Competition Question

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!

29 Sep 2013

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]

28 Sep 2013

APMO 1989 Q3: Geometry: Upper Secondary Mathematics Competition Question

Let A1, A2, A3 be three points in the plane, and for convenience, let A4 = A1, A5 = A2. For n = 1, 2, and 3, suppose that Bn is the midpoint of AnAn+1, and suppose that Cn is the midpoint of AnBn. Suppose that AnCn+1 and BnAn+2 meet at Dn, and that AnBn+1 and CnAn+2 meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3.

[APMO 1989 Q 3]

27 Sep 2013

Prize Maths Quiz: Integral Inscribed Triangles (PMQ38)

The triangle ABC has lengths AB = 60, AC = 63 and BC = 39. The point X lies on the circumcircle of ABC such that the triangle AXB also has all integral sides.

Find the perimeters of all possible integral triangles AXB.

Note that from next week, the PMQ will be renamed Professor Pailyn's Mathematical Quiz - so still a PMQ!

26 Sep 2013

OEMO Beginners 2001 Q4: Middle Secondary Mathematics Competition Question

Let ABC be a triangle whose angles α = CAB and β = CBA are greater than 45°.

Above the side AB we construct a right-angled isosceles triangle ABR with AB as hypotenuse, such that R lies inside the triangle ABC.

Analogously we erect above the sides BC and AC right-angled isosceles triangles CBP and ACQ, but with their (right-angled) vertices P and Q outside of the triangle ABC.

Show that CQRP is a parallelogram.

[OEMO Beginners 2001 Q4]

25 Sep 2013

OEMO Beginners 2001 Q2: Middle Secondary Mathematics Competition Question

We consider the quadratic equation x2 - 2mx - 1 = 0, where m is an arbitrary real number.

For which values of m does the equation have two real solutions, such that the sum of their cubes equals eight times their sum.

[OEMO Beginners 2001 Q2]

22 Sep 2013

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]

JBMO 2013 Q3: Inequality: Middle Secondary Mathematics Competition Question

Show that

for all positive real numbers a and b such that ab ≥ 1.

[Junior Balkan MO 2013 Problem 3][The original paper has 4 problems to be done in 4.5 hours]

OEMO Beginners 2000 Q2: Middle Secondary Mathematics Competition Question

Let a and b be positive real numbers. Prove that the inequality


When does equality hold?

[OEMO Beginners 2000 Q2]

21 Sep 2013

Prize Maths Quiz: Inscribed Hexagon Puzzle (PMQ37)

The figure ABCDEF is a cyclic hexagon with lengths AB = CD = EF = 5, BC = DE = 2 and AF = 11. Given that the area of the hexagon is 54, find the rational value of the length BE.

As usual, assume this is a competition question. You have an ample supply of paper and pens and... that's it! Of course, this isn't a timed competition so that finding the answer is not so hard, but the important thing is to have a good solution method that can also be used in the future. Enjoy!

OEMO Beginners 2000 Q3: Middle Secondary Mathematics Competition Question

A "nice" two-digit number is at the same time a multiple of the product of its digits and a multiple of the sum of its digits.

How many such two-digit numbers exist?

What is the quotient of number and sum of digits for each of these numbers?

[OEMO Beginners 2000 Q3]

19 Sep 2013

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.]

OEMO Beginners 2000 Q4: Middle Secondary Mathematics Competition Question

Let ABCDEFG be one half of a regular dodecagon.

Let P be the intersection of the lines AB and GF and let Q be the intersection of the lines AC and GE.

Show that Q is the circumcenter of the triangle AGP.

[OEMO Beginners 2000 Q4]

17 Sep 2013

OEMO Beginners 2013 Q4: Middle Secondary Mathematics Competition Question

Let ABC be an acute triangle and D be a point on the altitude through C.

Prove that the mid-points of the line segments AD, BD, BC and AC form a rectangle.

[OEMO Beginners 2013 Q 4, by G. Anegg, Innsbruck]

14 Sep 2013

Prize Maths Quiz: Intersecting Chords Game (PMQ36)

Imagine a circle with six distinct points marked on its circumference and labelled 1 to 6. There are also six cards numbered 1 to 6. The cards are shuffled and placed face down on a table. You pick two cards and look at the numbers on them; join the two points shown on the cards by a straight-line chord. You then pick two more cards and join those two points with a chord. Finally, a chord is drawn joining the last remaining pair of points.

You win the game if none of the chords intersect. What is the probability of winning this game?

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]

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