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Golden Construction

Age 16 to 18Challenge Level

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

Xtra

Age 14 to 18Challenge 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.

Kissing

Age 16 to 18Challenge Level

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Partly Circles

Age 14 to 16Challenge Level

What is the same and what is different about these circle questions? What connections can you make?

Pent

Age 14 to 18Challenge 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.

Darts and Kites

Age 14 to 16Challenge Level

Explore the geometry of these dart and kite shapes!

Pentakite

Age 14 to 18Challenge Level

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Golden Thoughts

Age 14 to 16Challenge Level

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A Third of the Area

Age 14 to 16 ShortChallenge Level

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?

Halving the Triangle

Age 16 to 18Challenge 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.

Golden Ratio

Age 16 to 18Challenge Level

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

Square Mean

Age 14 to 16Challenge Level

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Pareq Calc

Age 14 to 16Challenge Level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Golden Fibs

Age 16 to 18Challenge Level

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

Golden Mathematics

Age 16 to 18

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

Good Approximations

Age 16 to 18Challenge Level

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Age 16 to 18 ShortChallenge Level

Can you solve this problem involving powers and quadratics?

Symmetrically So

Age 16 to 18Challenge Level

Exploit the symmetry and turn this quartic into a quadratic.

How Many Balls?

Age 16 to 18Challenge 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.

Age 16 to 18Challenge Level

What do you get when you raise a quadratic to the power of a quadratic?

Age 16 to 18Challenge Level

Can you find a quadratic equation which passes close to these points?

Implicitly

Age 16 to 18Challenge Level

Can you find the maximum value of the curve defined by this expression?

Target Six

Age 16 to 18Challenge Level

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Interactive Number Patterns

Age 14 to 16Challenge Level

How good are you at finding the formula for a number pattern ?

Plus or Minus

Age 16 to 18Challenge Level

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

Polar Flower

Age 16 to 18Challenge Level

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

Always Two

Age 14 to 18Challenge 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.

Golden Eggs

Age 16 to 18Challenge Level

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

Resistance

Age 16 to 18Challenge Level

Find the equation from which to calculate the resistance of an infinite network of resistances.

How Old Am I?

Age 14 to 16Challenge Level

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Age 16 to 18Challenge Level

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Cocked Hat

Age 16 to 18Challenge Level

Sketch the graphs for this implicitly defined family of functions.

Continued Fractions II

Age 16 to 18

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

In Between

Age 16 to 18Challenge Level

Can you find the solution to this algebraic inequality?

Two Cubes

Age 14 to 16Challenge Level

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .