Quadratic equations

problem
Plus or minus

Age

16 to 18

Challenge level

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
problem
Golden eggs

Age

16 to 18

Challenge level

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
problem
Golden construction

Age

16 to 18

Challenge level

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.
problem
Polar flower

Age

16 to 18

Challenge level

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.
problem
Golden fibs

Age

16 to 18

Challenge level

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!
problem
In between

Age

16 to 18

Challenge level

Can you find the solution to this algebraic inequality?
problem
Xtra

Age

14 to 18

Challenge level

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.
problem
Halving the triangle

Age

16 to 18

Challenge level

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.
problem
Pentakite

Age

14 to 18

Challenge level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?
problem
How many balls?

Age

16 to 18

Challenge level

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.