Diophantine equations

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

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: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
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...
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

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

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