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=x²+2x−1 and y=x²x+1 split the plane into four regions:

parabolas splitting the plane into regions

What is the maximum number of regions that two parabolas of the form y=ax²+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.

Equilateral triangle

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

Square and pentagon

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 am where 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, p2+q2+r2, in n different ways (where the order doesn't matter). What is the value of n?

December 2018
Christmas trees

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: t1, t2, t3, ... and u1, u2, u3, ... are multiplied term-by-term to form the terms of a new sequence:
t1u1, t2u2, t3u3, ...

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 logo

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

Question 2

In this question a > 0.
The line y = 3ax and the curve y = x2 + 2a2 enclose an area of size a.
What is the value of a?

Question 3

Let f(x) = 10x2 + 100x + 10.

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

Given that ab, 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 x0, x1, x2, x3, ... is defined as follows:
x0=1, x2n+1=cos(x2n), x2n+2=arctan(x2n+1) for all integers n≥=0.

Find the limit to which the sequence yn = x2n+1x2n+2 (n≥=0) converges.

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

September 2018
Spiral of Theodorus

Spiral of Theodoru

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?

Spiral of Theodoru

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?

Square on a cubic

For which values of k is it possible to draw a square using four points that lie on the curve with equation y = x3kx?

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.

rectangle and square from a circle

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)

rectangle and square from a circle

May 2018
Rich Tasks for Further Maths

The loci for |z–(a+bi)|=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.

complex loci

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

egg construction

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?