Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

You add 1 to the golden ratio to get its square. How do you find higher powers?

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

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

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

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.

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

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

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

What are the possible remainders when the 100-th power of an integer is divided by 125?

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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

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?

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

What is the smallest perfect square that ends with the four digits 9009?

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

The sum of the cubes of two numbers is 7163. What are these numbers?

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.