Challenge Level

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

Challenge Level

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.

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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.

Challenge Level

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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.

Challenge Level

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

Challenge Level

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.

Challenge Level

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}.

Challenge Level

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

Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Challenge Level

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

Challenge Level

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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

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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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To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

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

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Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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Can you find a rule which connects consecutive triangular numbers?

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

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

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

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

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

Challenge Level

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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Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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Find the five distinct digits N, R, I, C and H in the following nomogram

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The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Challenge Level

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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However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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

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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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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 ?

Challenge Level

Can you prove that twice the sum of two squares always gives the sum of two squares?