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?

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?

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!**

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).

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*?

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 A(*ab*)=A(*a*)+A(*b*).

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.

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?

Star Cores

In a square with side length 1 the vertices are joined to the midpoints of the non-adjacent sides to make a star.

What is the area of the octagon created in the centre?

Quarter Master

Given an equilateral triangle, in how many different ways can you construct a shape that has an area that is a quarter of the original triangle, using just a straight edge and a pair of compasses?

An interactive version of this problem is available at: www.geogebra.org/m/ravcumBt.

Happy 2018

This New Year crosses over two years that can be written as the sum of the squares of two positive integers: 2017=9^{2}+44^{2} and 2018=13^{2}+43^{2}.

Will there ever be three years in a row that can be written as the sum of the squares of two positive integers?

Will there ever be four years in a row that can be written as the sum of the squares of two positive integers?