Diophantine equations

problem
Lattice points

Age

16 to 18

Challenge level

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?
problem
Cakes and buns

Age

11 to 14

Challenge level

Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?
problem
Are you kidding

Age

14 to 16

Challenge level

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?
problem
Letter land

Age

11 to 14

Challenge level

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.
problem
CD Heaven

Age

14 to 16

Challenge level

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at each price?
problem
Deep roots

Age

14 to 16

Challenge level

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
problem
Plutarch's boxes

Age

11 to 14

Challenge level

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?
problem
In particular

Age

14 to 16

Challenge level

Can you find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers?
problem
Whole numbers only

Age

11 to 14

Challenge level

Can you work out how many of each kind of pencil this student bought?
problem
Not a polite question

Age

11 to 14

Challenge level

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...