Problems

Let $n\ge r$ be positive integers. What is $F_n^2-F_{n-r}{F}_{n+r}$ in terms of $F_r$?

Have you wondered if $F_{-5}$ is possible? Here is how we can extend the Fibonacci sequence to the negative indices. The relation $F_{n+1}=F_n+F_{n-1}$ can be rewritten as $F_{n-1}=F_{n+1}-F_n$. We can simply define the Fibonacci sequence with negative indices with this formula. For example, $F_{-1}=F_1-F_0=1-0=1$.

Write out $F_{-1},F_{-2},\dots ,F_{-10}$. 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 $n\ge 1$, $$\sum _{k=1}^{n}s(k)=s(1)+s(2)+...+s(n)\le \frac{{\pi }^2}{12}{n}^2+\frac{n\log n}{2}+\frac{n}{2}.$$

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 $\text{mathsterling}1$ each. Golf clubs cost $\text{mathsterling}10$ each. At this shop, Ross purchased $100$ items for a total cost of exactly $\text{mathsterling}100$ (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 $a\otimes b$ with the following three properties.
1. $a\otimes a=a+2$;
2. $a\otimes b=b\otimes a$;
3. $\frac{a\otimes (a+b)}{a\otimes b}=\frac{a+b}{b}.$
Calculate $8\otimes 5$.

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 $$ATE\times EAT\times TEA=36239651.$$ What is $A\times E\times T$?

$x$, $y$ and $z$ are all integers. We’re told that $$\begin{array}{rl}{x}^{3}yz& =6\\ x{y}^{3}z& =24\\ xy{z}^3& =54.\end{array}$$ What’s $xyz$?

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove that $a^a+b^b+c^c\ge 3$.

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.