Products of Integer Compositions: Lower Secondary Mathematics Competition Question

The partition of a natural number is a way of writing the number as the sum of positive integers. For example, (1+2) is a partition of 3. If the order of the numbers in the sum is important we call this a composition; if we only include unique sums then that is strictly a partition.

So, the number 3 can be composed into 4 sums: 3, 1+2, 2+1 and 1+1+1. However, the unique partitions of number 3 are just 3, 2+1 and 1+1+1. Note that the partition with parts 2 and 1 could be written in one of two ways; the convention is to order such sums with the highest numbers first. Although this is not vitally important, it is good practice so that it is easier to see that you have not missed out on any partitions - and that you haven't duplicated any.

Now, let us go back to the compositions of 3 and we shall replace the summation sign with a multiplication sign. We therefore get the following list:
3
2x1=2
1x2=2
1x1x1=1
The sum of these products is 3+2+2+1=8.

Now, repeat the above process for numbers 5 and 6. That is, list all the integer compositions of 5, replace the summation signs with multiplication signs, and finally add together all these products. Then do the same for number 6.

You now have three sums of the products of the compositions of 3, 5 and 6. What do you think this total will be for the numbers 4 and 7?

What do you think this total will be for the number 15?





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All the Grids of Sums: Upper Secondary Mathematics Competition Question

The diagram below is the same as that used in the Grid of Sums question. However, in this case, I would like you to find all the possible solutions.

To restate the rules, the number in each circle is the sum of the numbers in the two squares above it. You must use every number from 1 to 9 only once.

List all the possible solutions. How many unique solutions are there? You may ignore reflections.

You will come across many questions such as this one, especially in any follow-up papers to open maths competitions. At first glance, there seem to be a huge number of options. Putting the nine numbers in randomly, there are 9!/2 permutations – too many to enumerate. However, the grid has some restrictions that will help you reduce the number of options dramatically.

This question does require enumeration, but you need to find a systematic way of doing it so that the answers can be found within a reasonable time. It also means that you can be confident of having found all the solutions.

Level: Upper Secondary (Red)


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Numbers in a Grid Puzzle: Middle Secondary Mathematics Competition Question

The grid below is to be filled with numbers, one in each cell and not necessarily integers.

The numbers are arranged in such a way that if you look at the number in one particular cell, the cell to the right is double that number, and the cell below it is half that number.

The four numbers at the corners of the grid add up to a positive integral cube number. What is the smallest possible value of the number in the middle of the grid?

Level: Middle Secondary (Yellow)



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A Grid of Sums: Lower Secondary Mathematics Competition Question

The diagram shows a row of five squares above a row of four circles. The number in each circle is generated by adding the numbers in the two squares immediately above it.

Using only the single-digit numbers 1 to 9, complete the grid so that every number is used once and every circle is the sum of the two squares above it. The diagram has the number 6 already inserted.

This question requires a bit of logic so that you can go through all the possible permutations in a systematic and quick way.

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A Squared Prime Problem: Upper Secondary Mathematics Competitions

There are prime numbers p, such that the number squared p² is a factor of [(p-1)! + 1].

For which of the following primes is this true?

2, 3, 5, 7, 11, 13, 17.

In a competition, without electronic devices, this question should take approximately 10 minutes. Doing the question by computer is trivial - although interesting as p becomes larger - the aim here is to find the fastest technique to do it manually, as if it were a test question.

You can post your answer in the Comments below, but please try it first!


Level: Upper Secondary (Red)



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Factorial Primes: Middle Secondary Mathematics Competitions

A factorial prime is a prime number of the form p = n! + 1 or p = n! – 1, where n is a positive integer.

Find the sum of all such distinct factorial primes under 50.

As always, show your solution. Add your answer to the Comments below.

In a competition, without any electronic devices, this question should take about 2 minutes.


Level: Middle Secondary (Yellow)



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Primes in a Sequence: Lower Secondary Mathematics Competition

We have a sequence of numbers defined as follows:

T(1) = 3, T(2) = 4, and T(n) = T(n-1) + T(n-2), where n is a positive whole number greater than 2.

T(1) is obviously a prime number. Find the next five prime numbers that appear in this sequence. Write down their sum.


Answer will eventually be posted below in the Comments section, but best to try it yourself first!


Level: Lower Secondary (Blue)

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Primeless Road: Upper Primary Mathematics Competition Question

Primeless Road is a rather peculiar road. All the odd numbered houses are on one side of the road, and all the even ones are on the other side, both sides starting with the lowest number. Nothing very strange with that! However, none of the house numbers is a prime number.

Assuming each house has exactly one other house directly opposite, what is the number of the house opposite number 20?

You can write your answer and discuss the question in the Comments section below.

Level: Upper Primary (Green)


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Midweek Maths: A Purse Full of Ducats

Three men are sitting round a table. They each have some 1-ducat coins in front of them and there is a purse full of similar coins in the middle of the table.


Alessandro turns to Bruno and says:”If I add the ducats in the purse to my money I will have twice as many ducats as you.” Then Bruno turns to Cesare and says:”If I add the coins in the purse to my money, I will have three times as many ducats as you.” Finally, Cesare says to Alessandro:”If I add the ducats in the purse to those I already have, I will have four times as much money as you.”

How many ducats does each man have and how many are in the purse? Assume the smallest integer solution.

As in the previous question, Fibonacci did not use any algebra to solve this question. For students who feel that algebra is some kind of mental torture, try following a verbal algorithm nearly a page long!

Math-e-Monday: A Merchant from Pisa

This week I am reading the Liber Abaci (The Book of Calculations) by Leonardo Pisano, better known as Fibonacci. The book, published in 1202, contains the famous problem of rabbits breeding in such a way as to generate the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 and so on. However, the Liber Abaci also contains a wealth of other problems, and this week I shall post a small selection of questions inspired by those found in the book.

The question below is fairly easy and can be done by a lower secondary, or even upper primary, student. Interestingly, Fibonacci did not have a well-developed algebra, so his solution is a numerical recipe or algorithm. As part of Fibonacci's promotion of the Hindu-Arabic numerals, he has numerous questions involving fractions. Many such questions are designed to show the ease with which fractions can be manipulated using his 'new' system of numbers compared with using Roman numerals. This is one such question.

Midweek Maths: Two Diamonds

We continue on the theme of geometry. Like the previous question, the one below can be done in different ways: with pure trigonometry; with geometric transformations; and even using coordinate geometry.

This is true of many questions in mathematics. An important skill in mathematics is not just the ability to solve problems, but the capacity to quickly go through different possible methods and to pick the most efficient. This is a skill that is difficult to teach but essential in being able to save time in a mathematics competition – time that will be precious for the more challenging questions. One way for you to measure the most efficient method is to actually do the question below in three different ways! Try it and see for yourself which method seems faster.

It is also important to develop some self-knowledge about which branches of mathematics you find come most naturally. Some students prefer number theory whereas others may like geometry or probabilities. In team competitions it is vital to distribute each given question to the person who is the ‘expert’ in that field. Sometimes, the best team is not always the sum of the best individuals.

Prize Maths Quiz: Winners of PMQ5

Thank you to everyone who submitted their answer to PMQ5 (Mercury and the Sum of a Series).

Before I announce the winners, I need to mention a slight change to how I do this from now on. I used to wait for the winners to respond to my congratulatory email so that I could list their names and countries. However, some don't bother replying, so that it looks as if nobody won! So, from now on, I shall announce the winners using just their initials or part of their email address. Then, once the winners have responded and the actual claimant student has accepted the prize, I will list the names as they join our online classroom.

Having got that out of the way, this week's winners are:

K.R.
M.K.
J.K.

An email has already been sent.

For the first time, we have had more submissions than prizes, so to those who didn't win this week, please try again next Friday.

Well done!

Richard
GiftedMathematics.com

Prize Maths Quiz: Trapezium Artist (PMQ4)

Alice was playing out in the garden. She really was getting far too big to be playing on the swing; this made her sad as she liked swinging up as high as she could. However, the creaking noises troubled her and her mind turned to whether she could unhinge the whole structure and fall head-first onto the lawn. It struck her as safer to do this as a theoretical calculation than as an experiment. She knew it could be done, but didn't yet know how.

“Alice! Come on, we’re going shopping!” bellowed her mother. Oh no... why don’t my parents shop on the internet and have it all delivered? Now that’s what supermarkets are for; otherwise they are just dull-markets. And why is my big brother never around when he could be useful? Alice jumped off the swing and rolled around on the grass, making herself look as unpresentable as possible.

“Come on, darling! You know I hate all that traffic later in the day.” Her mother fussed trying to brush off clumps of dried grass from Alice’s dress. That didn’t work!

Shopping malls are ghastly plastic places with very little of interest for Alice. She usually tries to head for the computer shops but they are always full of tall smelly teenagers staring at games. “Alice, I know you love to wander off and I’m not wasting my time looking for you. If we get separated I’ll wait for you in this coffee shop. OK?” Alice was not really old enough to be let out on her own, but she was good at wandering off on her own.

Math-e-Monday: Half a Triangle

Most of the questions posted so far have been about numbers, so this week we’re going to concentrate on geometry.

The people who set geometry questions in mathematics competition papers like to mess with your eyes! They are not really optical illusions but, rather like a good magician, you can’t see the whole picture because you’re concentrating only on what you’re being shown. Geometric diagrams do usually show you everything you need but not always everything you know. My advice is to redraw the diagram and fill in everything you know that is related to the question; it could be trigonometry or circle theorems or construction techniques. Then make sure you read the question carefully to check for details that are written down but not illustrated in the given diagram.

Lastly, brush up on some of the more obscure geometric theorems. You can always prove things from first principles, but in a maths test that wastes precious time. It is better to be prepared with a basket full of theorems and then figure out which ones you need.

The Question

The diagram above shows a circle centre C (not drawn to scale). The line CQ is perpendicular to the diameter PR. The triangle PQR has double the area of triangle PSR. Find the angle PRS.

This is not a PMQ so feel free to discuss this puzzle in the comments section below.

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Prize Maths Quiz (PMQ3) Announcement for Friday 25 January

The next Gifted Mathematics Prize Maths Quiz shall be PMQ3 and will be posted on Friday 25 January 2013 at 07:03 GMT. The competition will close, as always at 23:59 GMT on the Sunday.

Please use our offical GMT clock in the right-hand column.

Please also read the rules, which have been recently updated and clarified, on our PMQ Prize page.

See You Then!




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