Challenge Level

Can you hit the target functions using a set of input functions and a little calculus and algebra?

Challenge Level

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Challenge Level

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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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

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

Challenge Level

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?

Challenge Level

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.

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

Derive an equation which describes satellite dynamics.

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 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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Relate these algebraic expressions to geometrical diagrams.

Challenge Level

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?

Challenge Level

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

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

Challenge Level

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

Challenge Level

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.

Challenge Level

Can you find the lap times of the two cyclists travelling at constant speeds?

Challenge Level

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Challenge Level

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

Challenge Level

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.

Challenge Level

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

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.

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

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?

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

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

Challenge Level

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.

Challenge Level

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.

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

Can you find a rule which connects consecutive triangular numbers?

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

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

Challenge Level

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Challenge Level

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Challenge Level

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Challenge Level

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Challenge Level

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?