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### Schools Mathematics Grand Challenge

PROBLEM 1

Mary recently won 1,000 Euro in a local lottery competition and wishes to invest it for 5 years (60 months). She has spoken to 6 banks and each of them has offered her a slightly different deal on interest payments. The offers from the six banks were:

• Bank 1: 30% interest every 60 months
• Bank 2: 15% interest every 30 months
• Bank 3: 10% interest every 20 months
• Bank 4: 7.5% interest every 15 months
• Bank 5: 6% interest every 12 months
• Bank 6: 5% interest every 10 months

For instance, if she invested with Bank 2 she would receive: 150 Euro interest after the first 30 months, giving her a total of 1150 Euro; after another 30 months (at the end of 60 months) she would receive 15% interest on her 1150 Euro, which is 172.50 Euro, giving her a total of 1150 + 172.50 = 1322.50 Euro at the end of the 60 months.

Which bank will pay Mary the most interest over the 60 month period?

Please give your answer as a number from 1 to 7 (inclusive). If you think all banks offer the same deal, give the number 7 as your answer, otherwise your answer should be the number of the bank offering the best deal.

PROBLEM 2

Currently there are 7 days in a week and 365 days in a standard (non-leap) year. This means that the number of weeks in a standard (non-leap) year is not a whole number: in fact it is approximately 52.1429. It is proposed to change the number of days in a week so that the following conditions are satisfied:

1. The number of weeks in a standard (non-leap) year is a whole number;
2. The number of days in a week is a whole number;
3. A week cannot contain just 1 day;
4. A week cannot be 365 days long.

How many different possibilities are there for the length of a week under this scheme?

PROBLEM 3

John, Sheila and Mark meet at the bus-stop on a winter's morning. John is wearing a blue jacket, Sheila is wearing a black jacket, and Mark is wearing a red jacket. One of the three friends says:

"I am wearing a blue hat, one of you is wearing a red hat, and the other is wearing a black hat. But none of us is wearing a hat that matches the colour of our jacket."

Sheila then replies to him:

"You're absolutely correct."

What colour hat is Mark wearing? If you think his hat is black, then submit 1 as your answer; if you think the hat is blue, then submit 2 as your answer; if you think the hat is red, then submit 3 as your answer.

NOTE: All of the three friends speak the truth.