# Maths item of the month

### August 2014

Cubic with Integer Points

Find a cubic equation, with three distinct roots, such that the roots and the stationary points all lie on points with integer coordinates.

If the coefficient of *x*^{3} is equal to 1, what is the cubic with this property where the sum of the absolute value of the roots is as small as possible?

### July 2014

Summer All Over

Take the set of the reciprocals of all the positive integers from 1 to *n*.

Show that the sum of the products of the possible subsets (including the complete set) is equal to *n*.

e.g. for *n* = 3:

### June 2014

APGP

An arithmetic progression (AP) is a sequence of numbers with a common difference between them: e.g. 3, 7, 11, 15, …

A geometric progression (GP) is a sequence of numbers with a common ratio between them: e.g. 2, 6, 18, 54, …

If the 1^{st}, 2^{nd} and 6^{th} terms of an AP form a GP what is the common ratio?

If the 1^{st}, 2^{nd} and n^{th} terms of an AP form a GP what is the common ratio?

### May 2014

How many circles?

Circles plotted on coordinate axes can be categorised based on the following criteria (ignoring the scales on the axes):

- Position of the centre
- Positions of the points of intersection with the
*x*-axis - Positions of the points of intersection with the
*y*-axis

For example the circles above can be categorised as follows:

**Circle A**

- Centre in the top-right quadrant
- One intersection with the positive
*x*-axis and one intersection with the negative*x*-axis - One intersection with the positive
*y*-axis and one intersection with the negative*y*-axis

**Circle B**

- Centre on the positive
*x*-axis - Two intersections with the positive
*x*-axis - No intersections with the
*y*-axis

**Circle C**

- Centre in the bottom-right quadrant
- One (repeated) intersection with the positive
*x*-axis - Two intersections with the negative
*y*-axis

Using these criteria how many different categories of circles are there?

### April 2014

Divisibility of consecutive integers

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

Prove that, for any positive integer *n*, *N* is divisible by 2^{n} 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

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:

### 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.

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 .

She gives the reason that there are 3 students to be chosen and the proportion of girls is

so she calculated

Is she correct?

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