# Maths item of the month

## Curriculum mapping

A list of Maths Items of the Month categorised by GCSE/A level topics can be seen at: Maths Items of the Month Curriculum mapping.

## Recent Maths Items of the Month

### March 2019

Regional differences

The parabolas y=*x*^{²}+2*x*−1 and y=*x*^{²}−*x*+1 split the plane into four regions:

What is the maximum number of regions that two parabolas of the form *y*=*ax*^{²}+*bx*+*c* can split the plane into?

What is the maximum number of regions that three parabolas of this form, or four parabolas of this form, can split the plane into?

### February 2019

Thinking equilaterally

Two equilateral triangles share a common vertex. Show that the lengths marked *a* and *b* are equal for any such arrangement.

Investigate the ratio *a*:*b* for other pairs of regular polygons.

### January 2019

Happy 2019

Some problems about the number 2019:

- The number 2019, its double 4038, and its triple 6057 contain all 10 digits. What is the next number with this property?
- The number 2019 can be written as the sum of the first
*n*perfect powers (integers of the form*a*where^{m}*a*>0 and*m*≥2). What is the value of*n*? - 2019 is the smallest number that can be written as the sum of the squares of 3 primes,
*p*^{2}+*q*^{2}+*r*^{2}, in*n*different ways (where the order doesn't matter). What is the value of*n*?

### December 2018

Christmas trees

A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.

What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?

### November 2018

Terms of engagement

Two arithmetic sequences: *t*_{1}, *t*_{2}, *t*_{3}, ... and *u*_{1}, *u*_{2}, *u*_{3}, ... are multiplied term-by-term to form the terms of a new sequence:

*t*_{1}*u*_{1}, *t*_{2}*u*_{2}, *t*_{3}*u*_{3}, ...

The first three terms of the new sequence are 360, 756 and 1260. What is the fourth term?

Given that all the terms of the sequences are positive integers what could the original sequences be?

### October 2018

Ritangle competition

Ritangle is a competition for teams of students of A level Mathematics, the International Baccalaureate and Scottish Highers: integralmaths.org/ritangle. The first five questions will be released on 1st, 8th, 15th, 22nd, 29th October 2018. Correct answers to these questions release a clue for the final question.

**Question 1**

How many 8 digit numbers are there that are both:

a) divisible by 18

and

b) such that every digit is a 1 or a 2 or a 3?

**Question 2**

In this question *a* > 0.

The line *y* = 3*ax* and the curve *y* = *x*^{2} + 2*a*^{2} enclose an area of size *a*.

What is the value of *a*?

**Question 3**

Let f(*x*) = 10*x*^{2} + 100*x* + 10.

Suppose f(*a*) = *b* and f(*b*) = *a*.

Given that *a* ≠ *b*, what is f(*a* + *b*)?

**Question 4**

In this question *a* and *b* are positive. A quadrilateral is formed by the points A, B, C and D where A is (*a*,0), B is (0,*b*), C is (-1/*b*,0) and D is (0,-1/*a*). ABCD is always a trapezium.

If *a*=11 what value of *b* minimises the area of trapezium ABCD?

**Question 5**

In this question angles are in radians. An infinite sequence *x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}, ... is defined as follows:

*x*_{0}=1, *x*_{2n+1}=cos(*x*_{2n}), *x*_{2n+2}=arctan(*x*_{2n+1}) for all integers *n*≥=0.

Find the limit to which the sequence *y*_{n} = *x*_{2n+1} − *x*_{2n+2} (*n*≥=0) converges.

**Please don’t share answers outside your team, others are having
fun finding them!**

### September 2018

Spiral of Theodorus

The Spiral of Theodorus is constructed using an isosceles right-angled triangle with shorter sides of one unit. Successive right-angled triangles are then constructed with the previous hypotenuse as the base and a height of one unit.

How many triangles can be constructed before one of the triangles overlaps an existing triangle?

Show that the ring created between each pair of triangles has the same area as the circle at the centre (with radius one unit).

### July 2018

Square cubes?

For which values of *k* is it possible to draw a square using four points that lie on the curve with equation *y* = *x*^{3} − *kx*?

### June 2018

MEI Conference 2018 – A couple of taster problems

The 2018 MEI Conference takes place at the University of Keele on 28-30 June. To see details of the conference and the wide variety of sessions on offer visit the conference website: conference.mei.org.uk

The following problems featured in the 2017 Conference sessions Squaring the circle and other shapes and The history of logarithms.

In the diagram below AB is a diameter of the circle with centre O. The point C lies on AB and the points D and E lie on the perpendicular to AB through C, with D being on the circle and CE=CB. Show that the area of the rectangle CAHE is equal to the area of the square CFGD.

In the graphs below the area under the curve *y*=1/*x* between *x*=1 and *x*=*a* is represented by A(*a*). Use the graphs to show geometrically that the A(*ab*)=A(*a*)+A(*b*)

### May 2018

Rich Tasks for Further Maths

The loci for |*z*–(*a*+*b*i)|=*r* are drawn in an Argand diagram for each of the cases where *a*, *b* and *r* take a distinct value of either 1, 2 or 4.

Show that the six points A, B, C, D, E and F where two of the loci touch, but don’t cross, all lie on a straight line. Will this be the case if *a*, *b* and *r* take all possible distinct values from *any* set of three different numbers?

This problem is from one of forty *Rich tasks for Further Maths* which will appear in Integral this term. For more information visit integralmaths.org.

### April 2018

Easter egg

An egg is constructed as follows:

- Draw a circle of radius 1 at a point A.
- Mark diametrically opposite points on the circumference, B and C. Draw the circle with centre B through C and the circle with centre C through B.
- Join the two points of intersection, D and E, of these two circles and find one of the points of intersection, F, of this line with the original circle.
- Draw the largest circle with centre F that sits inside the circles with centres B and C.
- Join the arcs CB, BG, GH and HC.

What is the area of egg?