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Resources tagged with Manipulating algebraic expressions/formulae similar to Sperner's Lemma:

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Quadratic Harmony

Stage: 5 Challenge Level: Challenge Level:1

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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Pair Squares

Stage: 5 Challenge Level: Challenge Level:1

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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Diverging

Stage: 5 Challenge Level: Challenge Level:1

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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Janine's Conjecture

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Look Before You Leap

Stage: 5 Challenge Level: Challenge Level:1

Relate these algebraic expressions to geometrical diagrams.

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Always Perfect

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Binomial

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Mechanical Integration

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

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?

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How Many?

Stage: 5 Challenge Level: Challenge Level:1

Find all the solutions to the this equation.

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Three Ways

Stage: 5 Challenge Level: Challenge Level:1

Given that x + y = -1 find the largest value of xy (a) by coordinate geometry (b) by calculus (c) by algebra.

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And So on - and on -and On

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Perfectly Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Telescoping Functions

Stage: 5

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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Polynomial Relations

Stage: 5 Challenge Level: Challenge Level:1

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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Never Prime

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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DOTS Division

Stage: 4 Challenge Level: Challenge Level:1

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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Cosines Rule

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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System Speak

Stage: 4 and 5 Challenge Level: Challenge Level:1

Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6

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Chocolate 2004

Stage: 4 Challenge Level: Challenge Level:1

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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Absurdity Again

Stage: 5 Challenge Level: Challenge Level:1

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

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Complex Partial Fractions

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Reciprocals

Stage: 5 Challenge Level: Challenge Level:1

Prove that for any positive numbers x1,x2,...,xn (x1 + x2 + ... + xn)([1/(x1)] + [1/(x2)] + ... + [1/(xn)]) is greater than or equal to n^2.

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2008 January Poster

Stage: 3 and 4 Challenge Level: Challenge Level:1

Pair Products

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' Tis Whole

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Back to Basics

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Magic Sums and Products

Stage: 3, 4 and 5

How to build your own magic squares.

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Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Sums of Pairs

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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There and Back

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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Sweeping Satellite

Stage: 5 Challenge Level: Challenge Level:1

Derive an equation which describes satellite dynamics.

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Calculus Countdown

Stage: 5 Challenge Level: Challenge Level:1

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

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Operating Machines

Stage: 5 Challenge Level: Challenge Level:1

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

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Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Fibonacci Factors

Stage: 5 Challenge Level: Challenge Level:1

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

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More Polynomial Equations

Stage: 5 Challenge Level: Challenge Level:1

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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Poly Fibs

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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Incircles

Stage: 5 Challenge Level: Challenge Level:1

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 ...?

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Ball Bearings

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Dating Made Easier

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If a sum invested gains 10% each year how long before it has doubled its value?

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Matchless

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Root to Poly

Stage: 4 Challenge Level: Challenge Level:1

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is 1+sqrt2+sqrt3.

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Sums of Squares

Stage: 5 Challenge Level: Challenge Level:1

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?

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Nicely Similar

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Really Mr. Bond

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

115^2 = (11 x 12)x 25, that is 13225 895^2 = (89 x 90)x 25, that is 801025 Can you explain what is happening and generalise?

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Cocked Hat

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A new solution to a Tough Nut problem. Aleksander has drawn graphs for members of the family of functions given by the implicit equation (x^2 + 2ay -a^2)^2 = y^2(a^2 - x^2) corresponding to different. . . .

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The Medieval Octagon

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Always Two

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.