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Have you wondered if F5 is possible? Here is how we can extend the Fibonacci sequence to the negative indices. The relation Fn+1=Fn+Fn1 can be rewritten as Fn1=Fn+1Fn. We can simply define the Fibonacci sequence with negative indices with this formula. For example, F1=F1F0=10=1.

Write out F1,F2,,F10. What do you notice about the Fibonacci sequence with negative indices?

Let n be a positive integer. We denote by s(n) the sum of the divisors of n. For example, the divisors of n=6 are 1, 2, 3 and 6, so s(6)=1+2+3+6=12. Prove that, for all n1, k=1ns(k)=s(1)+s(2)+...+s(n)π212n2+nlogn2+n2.

A shop sells golf balls, golf clubs and golf hats. Golf balls can be purchased at a rate of 25 pennies for two balls. Golf hats cost \mathsterling1 each. Golf clubs cost \mathsterling10 each. At this shop, Ross purchased 100 items for a total cost of exactly \mathsterling100 (Ross purchased at least one of each type of item). How many golf hats did Ross purchase?

For every pair of integers a, b, we define an operator ab with the following three properties.
1. aa=a+2;
2. ab=ba;
3. a(a+b)ab=a+bb.
Calculate 85.

During a tournament with six players, each player plays a match against each other player. At each match there is a winner; ties do not occur. A journalist asks five of the six players how many matches each of them has won. The answers given are 4, 3, 2, 2 and 2. How many matches have been won by the sixth player?

The letters A, E and T each represent different digits from 0 to 9 inclusive. We are told that ATE×EAT×TEA=36239651. What is A×E×T?

x, y and z are all integers. We’re told that x3yz=6xy3z=24xyz3=54. What’s xyz?

Let a, b and c be positive real numbers such that a+b+c=3. Prove that aa+bb+cc3.

Find all functions f from the real numbers to the real numbers such that xy=f(x)f(y)f(x+y) for all real numbers x and y.