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

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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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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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Can you find a rule which connects consecutive triangular numbers?

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

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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 a rule which relates triangular numbers to square numbers?

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

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

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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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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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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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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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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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For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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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.

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

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

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. 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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If you know the perimeter of a right angled triangle, what can you say about the area?

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

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A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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Can you explain what is going on in these puzzling number tricks?

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

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Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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

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Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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

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

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By proving these particular identities, prove the existence of general cases.

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