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Resources tagged with Algebra - generally similar to Weekly Problem 33 - 2008:

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Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

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Broad Topics > Algebra > Algebra - generally

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Euler's Squares

Stage: 4 Challenge Level: Challenge Level:1

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square. Three of the numbers that he found are a = 18530, b=65570, c=45986. Find the fourth number, x. You. . . .

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Diophantine N-tuples

Stage: 4 Challenge Level: Challenge Level:1

Take any whole number q. Calculate q^2 - 1. Factorize q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .

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Our Ages

Stage: 4 Challenge Level: Challenge Level:1

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .

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Loopy

Stage: 4 Challenge Level: Challenge Level:1

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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Converse

Stage: 4 Challenge Level: Challenge Level:1

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .

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Interactive Workout - Mathmo

Stage: 5 Short Challenge Level: Challenge Level:1

Mathmo is a revision tool for post-16 mathematics. It's great installed as a smartphone app, but it works well in pads and desktops and notebooks too. Give yourself a mathematical workout!

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Coffee

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

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Power Quady

Stage: 5 Challenge Level: Challenge Level:1

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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Novemberish

Stage: 4 Challenge Level: Challenge Level:1

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

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DOTS Division

Stage: 4 Challenge Level: Challenge Level:1

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Bang's Theorem

Stage: 4 Challenge Level: Challenge Level:1

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

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Rudolff's Problem

Stage: 4 Challenge Level: Challenge Level:1

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

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Medallions

Stage: 4 Challenge Level: Challenge Level:1

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized. . . .

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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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Interactive Workout - Further

Stage: 5 Challenge Level: Challenge Level:1

Give your further pure mathematics skills a workout with this interactive and reusable set of activities.

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Reaction Rates!

Stage: 5

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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The Development of Algebra - 1

Stage: 3, 4 and 5

This is the first of a two part series of articles on the history of Algebra from about 2000 BCE to about 1000 CE.

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For What?

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Gassy Information

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Do each of these scenarios allow you fully to deduce the required facts about the reactants?

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Diamonds Aren't Forever

Stage: 5 Challenge Level: Challenge Level:1

Ever wondered what it would be like to vaporise a diamond? Find out inside...

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Bird-brained

Stage: 5 Challenge Level: Challenge Level:1

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.