Reasoning, convincing and proving

problem
Even so

Age

11 to 14

Challenge level

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
problem
Tis unique

Age

11 to 14

Challenge level

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
problem
Pent

Age

14 to 18

Challenge level

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
$Unit fractions$

problem
Unit fractions

Age

11 to 14

Challenge level

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
problem
Aba

Age

11 to 14

Challenge level

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.
problem
American billions

Age

11 to 14

Challenge level

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
problem
Chocolate maths

Age

11 to 14

Challenge level

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it works?
problem
Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
problem
Parabella

Age

16 to 18

Challenge level

This is a beautiful result involving a parabola and parallels.
problem
AMGM

Age

14 to 16

Challenge level

Can you use the diagram to prove the AM-GM inequality?