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#### Resources tagged with Mathematical reasoning & proof similar to Reciprocals:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

### Perfectly Square

##### Stage: 4 Challenge Level:

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

### 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.

### Common Divisor

##### Stage: 4 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

### 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. . . .

### Leonardo's Problem

##### Stage: 4 and 5 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?

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

### 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.

### 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.

### A Biggy

##### Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

### Three Ways

##### Stage: 5 Challenge Level:

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

### Big, Bigger, Biggest

##### Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

### 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?

### Pythagoras Proofs

##### Stage: 4 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?

##### Stage: 4 Challenge Level:

Four jewellers share their stock. Can you work out the relative values of their gems?

### AMGM

##### Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

### Archimedes and Numerical Roots

##### Stage: 4 Challenge Level:

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?

### Geometric Parabola

##### Stage: 4 Challenge Level:

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

### 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.

### 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.

### Mechanical Integration

##### Stage: 5 Challenge Level:

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

### Mediant

##### Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### Number Rules - OK

##### Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

### Composite Notions

##### Stage: 4 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

### Unit Interval

##### Stage: 4 and 5 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?

### Modulus Arithmetic and a Solution to Dirisibly Yours

##### Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

### More Sums of Squares

##### Stage: 5

Tom writes about expressing numbers as the sums of three squares.

##### 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.

### Little and Large

##### Stage: 5 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?

### Look Before You Leap

##### Stage: 5 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

### Sums of Squares and Sums of Cubes

##### Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

### More Number Pyramids

##### Stage: 3 and 4 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

##### Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Without Calculus

##### Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

### 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.

### 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.

### Fractional Calculus III

##### Stage: 5

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

### The Triangle Game

##### Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

### Proofs with Pictures

##### Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### Impossible Sandwiches

##### Stage: 3, 4 and 5

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

### Euclid's Algorithm II

##### Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

### Diophantine N-tuples

##### Stage: 4 Challenge Level:

Can you explain why a sequence of operations always gives you perfect squares?

### Continued Fractions II

##### Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

### A Computer Program to Find Magic Squares

##### Stage: 5

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

### Mouhefanggai

##### Stage: 4

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

##### Stage: 5 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

### Circle Box

##### Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

### Pareq Exists

##### Stage: 4 Challenge Level:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

### Square Pair Circles

##### Stage: 5 Challenge Level:

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

### Pythagorean Golden Means

##### Stage: 5 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

### Euler's Squares

##### Stage: 4 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...