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If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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Can you find the value of this function involving algebraic fractions for x=2000?

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Relate these algebraic expressions to geometrical diagrams.

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Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

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If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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If you know the perimeter of a right angled triangle, what can you say about the area?

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How good are you at finding the formula for a number pattern ?

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A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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Show that all pentagonal numbers are one third of a triangular number.

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Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Can you find a rule which relates triangular numbers to square numbers?

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Can you find a rule which connects consecutive triangular numbers?

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Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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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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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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The sums of the squares of three related numbers is also a perfect square - can you explain why?

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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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Can you explain why a sequence of operations always gives you perfect squares?

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Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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How many winning lines can you make in a three-dimensional version of noughts and crosses?

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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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Five equations... five unknowns... can you solve the system?

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Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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There are unexpected discoveries to be made about square numbers...

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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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Kyle and his teacher disagree about his test score - who is right?

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What is special about the difference between squares of numbers adjacent to multiples of three?

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A task which depends on members of the group noticing the needs of others and responding.

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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?