Explaining, convincing and proving

problem

Arrowhead

Age

14 to 16

Challenge level

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
problem

Pareq Exists

Age

14 to 16

Challenge level

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
problem

Three Balls

Age

14 to 16

Challenge level

Do points P and Q lie inside, on, or outside this circle?
problem

Folding Squares

Age

14 to 16

Challenge level

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
problem

Rhombus in Rectangle

Age

14 to 16

Challenge level

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
problem

Matter of Scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?
problem

Mediant Madness

Age

14 to 16

Challenge level

Kyle and his teacher disagree about his test score - who is right?
problem

Archimedes and Numerical Roots

Age

14 to 16

Challenge level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
problem

Ordered Sums

Age

14 to 16

Challenge level

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.
problem

Day of the Triffids

Age

14 to 16

Challenge level

Jasmine buys three different types of plant. How many triffids did she buy?