Choosing crates with Santa

Santa is choosing which of two crates to use on his sled. They are both 5m high and both have bases with diagonals 10m. One of them has a square base and the other has an oblong base.

Santa wants to use the crate with the largest volume. Which one should he choose and why?

A prime example

Given the list of the first *n* primes, 2,3,5,..., the product of the primes plus one will be coprime to the list of primes used.

e.g.

*E*_{1} = 2 + 1

= 3

and 3 is coprime to 2.

*E*_{2} = 2×3 + 1

= 7

and 7 is coprime to 2 and 3.

Will *E _{n}* always be a prime number?

Will *E _{n}* ever be a square number?

A cute result

For an acute-angled triangle is the following statement always true?

sin*A* + sin*B* + sin*C* > 2

Easy as 1, 2, 3?

ABCD is a square. Find the angle labelled alpha.

Tangents to a Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on a circle. ABCD is a cyclic quadrilateral. Tangents to the circle (centre O) are drawn at each of A, B, C and D. The tangents at neighbouring vertices intersect at E, F, G and H to give a quadrilateral. Under what circumstances is EFGH cyclic?

Fair share

The numbers 1-10 are split into two groups of 5 such that

*a*_{1} > *a*_{2} > *a*_{3} > *a*_{4} > *a*_{5}

and

*b*_{1} < *b*_{2} < *b*_{3} < *b*_{4} < *b*_{5}.

Show that, for any split, |*a*_{1} − *b*_{1}| + |*a*_{2} − *b*_{2}| + |*a*_{3} − *b*_{3}| + |*a*_{4} − *b*_{4}| + |*a*_{5} − *b*_{5}| = 5².

e.g.

10 > 7 > 6 > 3 > 1

2 < 4 < 5 < 8 < 9

|10 − 2| + |7 − 4| + |6 − 5| + |3 − 8| + |1 − 9| = 25

Will this work for any list of numbers of size 2*n*?

i.e.

|*a*_{1} − *b*_{1}| + |*a*_{2} − *b*_{2}| + ... + |*a _{n}* −

Splitting a field

How can an irregular pentagonal field be equally divided between 2 farmers if the field is convex?

If the field were not convex, would it still be possible?

This problem is taken from a 2012 MEI Conference session. For details of the 2013 MEI conference see: conference.mei.org.uk.

The two colour theorem?

Given *n* circles in the plane, prove that no matter how these circles are arranged, the map which they form may be properly coloured with two colours.

This problem is taken from a 2012 MEI Conference session. For details of the 2013 MEI conference see: conference.mei.org.uk.

April fool: 2+2=5

Can you spot the mistake in the following "proof" that 2+2=5?

Next factor

Starting with the number 2, integers are added, in order, as vertices to a graph so that any two vertices are joined if one of them is a factor of the other. A graph for {2,3,4,5,6} is shown.

What is the maximum integer that can be reached if none of the edges of the graph are allowed to cross?

This is a version of the nrich problem Factors and Multiples Graphs.

For more problems see the nrich website.

Squaring the circle

In the diagram below the circle and square have the same centre and the same perimeter.

What fraction of the area of square lies outside the circle?

What fraction of the area of circle lies outside the square?

Trying Triangulars

The triangular numbers are 1, 3, 6, 10, ...

17 can be expressed as the sum of distinct triangular numbers:

17 = 1 + 6 + 10.

5 cannot be expressed as the sum of distinct triangular numbers.

What is the largest positive integer that cannot be expressed as the sum of distinct triangular numbers?