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.

Challenge Level

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Challenge Level

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

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

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

Challenge Level

An inequality involving integrals of squares of functions.

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

Which of these continued fractions is bigger and why?

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?

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

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

Challenge Level

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.

Challenge Level

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?

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

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

Challenge Level

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Challenge Level

Use the diagram to investigate the classical Pythagorean means.

Challenge Level

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

Challenge Level

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

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

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?

Challenge Level

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

Challenge Level

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

Challenge Level

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Challenge Level

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?

Challenge Level

Can you build a distribution with the maximum theoretical spread?

Challenge Level

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

Challenge Level

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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.

Challenge Level

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?