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In Particular

Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Then find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers.

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For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Training Schedule

Stage: 4 Challenge Level: Challenge Level:1

Why not use a spreadsheet to calculate quickly the score for each event given the athlete's times or distances?
Then try tweaking her times and distances to see how her scores can be affected by different training schedules.