January 2002, Stage 4&5

Problems

Get Cross

Stage: 4 Challenge Level:

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Matter of Scale

Stage: 4 Challenge Level:

Prove Pythagoras Theorem using enlargements and scale factors.

Deep Roots

Stage: 4 Challenge Level:

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

Sixational

Stage: 4 and 5 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Proof Sorter - Sum of an AP

Stage: 5 Challenge Level:

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

Big, Bigger, Biggest

Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

In Between

Stage: 5 Challenge Level:

Can you find the solution to this algebraic inequality?