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, 2017.** 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. The problems can also be downloaded from here.

**1. **Suppose that alloy A contains 40% gold by weight, and alloy B contains 25% gold. How much of each of these alloys should be taken to be melted together so as to produce 600 grams of alloy containing 30% gold?

**2. **In a rectangle , the perpendicular dropped from the vertex onto the diagonal divides in the ratio . Find the length of given the length .

**3. **Solve the inequality .

**4. **Given an arbitrary convex quadrangle , each side of it is divided by two points in the ratio . Then a new quadrangle is formed by constructing straight lines through pairs of these points as shown in the picture. Prove that the areas of the quadrangles and are equal.**5. **Prove that, given any 11 integers, one can choose some of them and put signs or between these chosen integers in such a way that the resulting sum is divisible by 2016.

(Recall that an integer is divisible by an integer if there is an integer such that ; in particular, is also divisible by .)

**6. **Given non-zero real numbers, one can perform the following operation: replace any two of these numbers and by the numbers and . Prove that one cannot return to the initial set of numbers after several such steps.

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.

Reblogged this on Algebra in Lincoln.

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