Problems

The AM-GM inequality asserts that the arithmetic mean of nonnegative numbers is always at least their geometric mean. That is, if $a_1,\dots ,a_n\ge 0$, then $$\frac{a_1+\cdots +a_n}{n}\ge \sqrt[n]{a_1\dots a_n}.$$ Prove this inequality.

There are many proofs of this fact and quite a few of them are by induction. In fact, one of the most creative uses of induction can be found in Cauchy’s proof of the AM-GM inequality in Cours d’analyse.

Consider the $4!$ possible permutations of the numbers $1,2,3,4$. Which of those permutations keep the expression $x_1{x}_2+x_3{x}_4$ the same?

Show that if $1+3+5+7+...+97+99={50}^2$, then $1+3+5+7+...+97+99+101={51}^2$. Don’t forget that $(a+b{)}^2=a^2+2ab+b^2$.

Prove that for all positive integers $n$ there exists a partition of the set of positive integers $k\le 2^{n+1}$ into sets $A$ and $B$ such that $$\sum _{x\in A}{x}^i=\sum _{x\in B}{x}^i$$ for all integers $0\le i\le n$.

Diophantine equations are those where we’re only interested in finding the integer solutions. Some of these equations are quite simple, while others look simple but are actually very difficult. The most famous one is Fermat’s Last Theorem, which says that when $n>2$, there are no solutions to $$x^n+y^n=z^n.$$ Pierre de Fermat claimed that he proved this in 1637, scribbling it in the margin of a book, but said “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." It was only proved by Andrew Wiles in 1995. Today’s problems won’t take 358 years to solve.

For which $n$ is the expression $4^n+n^4$ prime?

Find all solutions to $x^2+2=y^3$ in the natural numbers.

McDonald’s used to sell Chicken McNuggets in boxes of 6, 9 or 20 in the UK before they introduced the Happy Meal. What is the largest number of Chicken McNuggets that cannot be bought? For example, you cannot buy 8 Chicken McNuggets, but you can buy $21=6+6+9$ Chicken McNuggets.

Show that the equation $x^4+y^4=z^4$ cannot satisfied by integers $x,y,z$ if none of them are 0.

A regular polygon has integer side lengths and its perimeter is 60. How many sides can it have?