this month, Stage 3 & 4

May 2001

 

Problems

problem Icon

Converging Means

Stage:3 Challenge Level:Challenge Level:1

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

problem Icon

The Genie in the Jar

Stage:3 Challenge Level:Challenge Level:1

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

problem Icon

Some Spirals and Tessellations

Stage:3 Challenge Level:Challenge Level:1

Aspiral is a curve formed by a point moving around a fixed point and constantly moving away from or approaching the fixed point. This month, building on the idea that many sets of things can be. . . .

problem Icon

Mathland Election

Stage:3 Challenge Level:Challenge Level:1

A political commentator summed up an election result. Given that there were just four candidates and that the figures quoted were exact find the number of votes polled for each candidate.

problem Icon

Euler's Squares

Stage:4 Challenge Level:Challenge Level:1

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square. Three of the numbers that he found are a = 18530, b=65570, c=45986. Find the fourth number, x. You. . . .

problem Icon

Diophantine N-tuples

Stage:4 Challenge Level:Challenge Level:1

Take any whole number q. Calculate q^2 - 1. Factorize q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .

problem Icon

Lawnmower

Stage:4 Challenge Level:Challenge Level:1

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of. . . .

problem Icon

Gnomon I

Stage:4 Challenge Level:Challenge Level:1

A gnomon is an L shape. Each Fibonacci number has its gnomon with the area corresponding to the number. Draw them and show that they obey the Fibonacci rule.

problem Icon

Base Ratios

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

A triangle is divided up into 3 smaller triangles and a quadrilateral. The areas of the smaller triangles are 5, 8 and 10 square units as shown. What is the area of the quadrilateral?

problem Icon

More Parabolic Patterns

Stage:4 and 5 Challenge Level:Challenge Level:1

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

problem Icon

Proof Sorter - Quadratic Equation

Stage:4 and 5 Challenge Level:Challenge Level:2Challenge Level:2

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.