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This problem required making logical arguments that come together to form a proof. It can be hard to put all the steps in the right place but we received great solutions from Michael from the UK, Kaan from the German Highschool, Istanbul, Turkey and Alex from England. They all used very similar ideas and Michael produced a very well written and formatted word document explaining the ideas. The full solution is in PDF format at the following link: Interpolating Polynomials Soln1.pdf 

Thanks for the fantastic ideas you sent in!