For a chance to win the first prize (£100 Amazon voucher), or one of our prizes for runners-up, submit your typed or neatly written solutions of the following problems to** maths@lincoln.ac.uk **or by post to **Mathematics Challenge, School of Mathematics and Physics, University of Lincoln, Lincoln, LN6 7TS.** Please include your full name, postal address and email, as well as the name and address of your school. **The closing date is 5 January, 2016.** The competition is open to all young pre-university people in UK aged 16–18 years. It is not open to current university students. See full Terms and Conditions.

**1.** Suppose that an underground tunnel of length 100 km is constructed strictly along a straight line between two points on the Earth’s surface. Assuming that the Earth is a perfect sphere of radius 6,371 km, estimate the maximum depth which the tunnel reaches below ground.

**2.** Find all solutions of the equation .

**3.** Prove that a number 2^{n }, where *n* is a positive integer, cannot have four equal right-most digits.

**4.** In the following picture there are three squares of side length 1. Find the area of the triangle *ABC*.

**5.** Prove that if *a,b,c* are odd integers, then the equation *ax*^{2} + *bx* + *c* = 0 has no rational roots.

**6.** Every square of a 2015 × 2015 table contains either 1 or –1. It is known that the sum of the numbers in every 2 × 2 square is equal to zero. Prove that the sum of all numbers in the table cannot be greater than 2015.

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 six problems – so you are encouraged to submit solutions even if you do only some of the problems.

[…] to the winners of the Mathematics Challenge 2015-16 and a big thank you to all participants. The Awards Ceremony has taken place on 18th of March […]

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