Diophantine equations

problem
Upsetting Pythagoras

Age

14 to 18

Challenge level

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2
problem
BT... eat your heart out

Age

16 to 18

Challenge level

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?
problem
Exhaustion

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?
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
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
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
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
Hallway borders

Age

11 to 14

Challenge level

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?
problem
Deep roots

Age

14 to 16

Challenge level

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$