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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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.

What fractions can you find between the square roots of 65 and 67?

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

Use the diagram to investigate the classical Pythagorean means.

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

The challenge is to find the values of the variables if you are to solve this Sudoku.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

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

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

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

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?

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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.

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.

An inequality involving integrals of squares of functions.

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Can you build a distribution with the maximum theoretical spread?

Can you prove our inequality holds for all values of x and y between 0 and 1?

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Can you find a quadratic equation which passes close to these points?

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Which of these continued fractions is bigger and why?

Find all the periodic cycles and fixed points in this number sequence using any whole number as a starting point.

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?