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Broad Topics > Algebra > Manipulating algebraic expressions/formulae

Triangles Within Squares

Stage: 4 Challenge Level:

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

Nicely Similar

Stage: 4 Challenge Level:

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Incircles

Stage: 5 Challenge Level:

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Ball Bearings

Stage: 5 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.

Salinon

Stage: 4 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?

The Medieval Octagon

Stage: 4 Challenge Level:

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Simplifying Doughnut

Stage: 4 and 5 Challenge Level:

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

Stage: 4 Challenge Level:

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

Interpolating Polynomials

Stage: 5 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.

Chocolate 2010

Stage: 4 Challenge Level:

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

Matchless

Stage: 4 Challenge Level:

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 ?

Always Two

Stage: 4 and 5 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.

Back to Basics

Stage: 4 Challenge Level:

Find b where 3723(base 10) = 123(base b).

Algebra Match

Stage: 3 and 4 Challenge Level:

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

Magic Sums and Products

Stage: 3 and 4

How to build your own magic squares.

Sums of Pairs

Stage: 3 and 4 Challenge Level:

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?”

Reciprocals

Stage: 5 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.

Sitting Pretty

Stage: 4 Challenge Level:

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?

Algebra from Geometry

Stage: 3 and 4 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.

Graphic Biology

Stage: 5 Challenge Level:

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

There and Back

Stage: 4 Challenge Level:

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

System Speak

Stage: 5 Challenge Level:

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

Lap Times

Stage: 4 Challenge Level:

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

Stage: 5 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.

Pair Squares

Stage: 5 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.

More Polynomial Equations

Stage: 5 Challenge Level:

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.

' Tis Whole

Stage: 4 and 5 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?

Never Prime

Stage: 4 Challenge Level:

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.

Complex Partial Fractions

Stage: 5 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.

And So on - and on -and On

Stage: 5 Challenge Level:

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

Janine's Conjecture

Stage: 4 Challenge Level:

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. . . .

Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level:

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

Calculus Countdown

Stage: 5 Challenge Level:

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

Cosines Rule

Stage: 4 Challenge Level:

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Absurdity Again

Stage: 5 Challenge Level:

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

Particularly General

Stage: 5 Challenge Level:

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

DOTS Division

Stage: 4 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}.

Sweeping Satellite

Stage: 5 Challenge Level:

Derive an equation which describes satellite dynamics.

Operating Machines

Stage: 5 Challenge Level:

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

Perfectly Square

Stage: 4 Challenge Level:

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

Diverging

Stage: 5 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.

Orbiting Billiard Balls

Stage: 4 Challenge Level:

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Always Perfect

Stage: 4 Challenge Level:

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

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

Polynomial Relations

Stage: 5 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.

Fibonacci Factors

Stage: 5 Challenge Level:

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

Poly Fibs

Stage: 5 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.

Root to Poly

Stage: 4 Challenge Level:

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Three Ways

Stage: 5 Challenge Level:

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

Sums of Squares

Stage: 5 Challenge Level:

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?