Maths item of the month

April 2014
Divisibility of consecutive integers

Let N = (n+1)(n+2)...(2n), i.e. the product of n consecutive integers from n+1 to 2n.

Prove that, for any positive integer n, N is divisible by 2n but not a higher power of 2.

For example, for n = 3:
N = 4×5×6
    = 120.
120 is divisible by 8 but not divisible by 16.

March 2014
Inner Circle

circles in regular polygons

In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C.

Show that for any regular polygon drawn in this way:

Area of P/Perimeter of P = Area of C/Circumference of C

February 2014
Cones from a Circle

An angle θ is cut out of a circle of card to create two sectors: a major sector and a minor sector. The two sectors are then folded to make cones.

Cones from a circle

What angle θ is required to obtain the largest value for the sum of the volumes of the two cones?

January 2014
An Unexpected Answer

Mr Student sets his class the following problem:

A committee of 3 students is to be chosen from a group of 13 students of which 8 are girls and 5 are boys. The students are selected at random, without replacement. What is the expected number of girls on the committee?

Anne Student immediately responds that the answer is 24/13 .
She gives the reason that there are 3 students to be chosen and the proportion of girls is 8/13
so she calculated 3*8/13=24/13

Is she correct?
If the number of students was different would her method work?