Mathematical Olympiad for Girls

24th February 2020

Divya Agrawal (Year 12) and Lydia Nottingham (Year 11) took part in this year's Mathematical Olympiad for Girls. Over 1400 girls nationwide participated, with top 25% receiving a Certificate of Distinction. The Mathematical Olympiad for Girls is an event run by the UK Mathematical Trust, introduced in 2011, to help schools nurture the talents of enthusiastic young female mathematicians. It is aimed at girls in Year 13 and below.

Lydia achieved a certificate of Distinction and Divya a certificate of participation, well done girls!

This was one of the questions on the paper. What would you score on it?

Let n be a positive integer. Tracy writes a list of 10 whole numbers between 1 and n (inclusive). Each number in the list is either equal to, one less than, or one more than the number before it. For example, when n = 7: Her list could be 5, 5, 6, 7, 6, 6, 5, 5, 6, 6 or 4, 4, 3, 2, 1, 1, 1, 1, 1, 1. Her list could not be 1, 3, 3, 4, 5, 5, 6, 7, 7, 7 or 5, 6, 7, 8, 7, 6, 5, 5, 5, 5.
(a) Suppose that n = 3. Stacey forms a list by copying Tracy’s list, except that whenever Tracy writes a 1, Stacey writes a 3, and whenever Tracy writes a 3, Stacey writes a 1.
(i) Which lists could Tracy write that would cause her list to be the same as Stacey’s?
(ii) Explain why Tracy can write as many lists that start 2, 2, 1 as start 2, 2, 3. (3 marks)
(b) For which n between 1 and 10 (inclusive) is the number of lists that Tracy could write odd? (7 marks)


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