# Mathematics Challenge 2014

This Competition has now closed. The winner will be notified by email and/or letter not later than 15 February 2015. If the winner cannot be contacted or does not claim the prize within 14 days of notification, we reserve the right to withdraw the prize from the winner and pick a replacement winner.

1. Estimate the distance from which the tower of Lincoln Cathedral appears the same size as the diameter of the Sun. Assume that the height of the tower is 83 m.
2. Find the right-most digit of the number 72014 (The 2014-th power of 7).
3. Find the right-most digit of the number 7(7 2015(7 to the power of 2015-th power of 7).
4. Given a square ABCD and a point O inside, there are two perpendicular lines through O. They intersect sides AB in P, BC in Q, CD in R, and DA in S. Thus, four quadrangles are formed: APOS, BQOP, CROQ, and DSOR. Prove that the sum of the perimeters of APOS and CROQ is equal to the sum of the perimeters of BQOP and DSOR.
5. How many sequences of length 10 can be composed of two letters A and B (in various proportions) such that no two letters B stand next to each other? (E.g. ABAABAAAAB is allowed but ABBAAAAAAA is not. You may use binomial coefficients to express your answer).

## Notes

Full solutions are required – not just answers – with complete proofs of any assertions you may make.

A winning submission may not necessarily be based on all five problems – so you are encouraged to submit solutions even if you do only some of the problems.