Reasoning, convincing and proving

problem
No right angle here

Age

14 to 16

Challenge level

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
problem
Always two

Age

14 to 18

Challenge level

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
problem
Number sandwiches

Age

7 to 14

Challenge level

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
problem
2-digit square

Age

14 to 16

Challenge level

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
problem
Two ladders

Age

14 to 16

Challenge level

Two ladders are propped up against facing walls. At what height do the ladders cross?
problem
Doesn't add up

Age

14 to 16

Challenge level

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
problem
Triangle midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
problem
Quadratic harmony

Age

16 to 18

Challenge level

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
problem
Mechanical integration

Age

16 to 18

Challenge level

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
problem
Loopy

Age

14 to 16

Challenge level

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?