# Winners of the Mathematics Challenge 2014

Congratulations to our winners and a big thank you to all those of you who entered.

## Joint 1st prize

Sylvia Su

Lincoln Minster Senior School, Lincoln

Laurie Wilkes

Beauchamp College, Leicester

### Joint 2nd prize

Hannah Blount

Derby College, Derby

Blessing Charumbira

The Priory Academy LSST, Lincoln

Charlie Hewis

Caistor Grammar School, Caistor

### Joint 3rd prize

Austin Hubbard

Southwell Minster School, Southwell

Jawaad Muhammad

Wyggeston & Queen Elizabeth I College, Leicester

Victoria Race

Kesteven & Grantham Girls School, Grantham

Phoebe Young

Queeen Elizabeth Grammar School, Alford

**Mathematics Challenge 2014 Solutions (PDF)**

# Mathematics Challenge 2014

## Give your brain a workout!

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 7^{2014}(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.