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# July/August 2015 - Summer Relay

Each question apart from the first involves in some way the answer to the previous question. Only the final answer will be accepted.

1: John and James are perfect logicians and both are always truthful. Once they met and James told John that one day he met a man with three children. John does not know this man or the ages of his children. James told John that the product of the children's ages was 72. John could not figure out the ages. James told John the sum of their ages but John could still not figure out the ages. What was the sum of the children's ages?

2: Let a be the answer to the previous question. It is given that x is a real number and y is an integer. Also, 5x2-6xy+2y2-2x+a=12. What is the value of y?

3: Let b be the answer to the previous question. Mr.A, Mr.B, Mr.C, and Mr.D are back. The following clues are given to their ages:

Mr.B is b years older than Mr.D.
23 years ago, Mr.B was twice as old as Mr.A.
24 years ago, Mr.C was twice as old as Mr.A.
When Mr.B was born, Mr.C was as old as Mr.A was when Mr.D was as old as Mr.B was when Mr.C was half as old as Mr.A will be in 16 years.

How old is Mr.D now?

4. Let c be the answer to the previous question. There is a c by c grid of unit squares. Five circles of radius 10 are placed on the grid, one centred on each corner and one centred on the centre of the grid. All unit squares which overlap with a circle are removed. The remaining squares are rearranged to form the largest square possible (with some squares missing). What is the side length of this square?

5. Let d be the answer to the previous questions. Let n and k be integers such that 1 ≤ k ≤ n. From the set of integers from 1 to n, the number k is removed. The average of the remaining numbers is d + 1/2.

The final answer is the sum of all possible values for n and k.