Mathematics

Start with any positive integer $n$. If $n$ is even, replace it by $n/2$, and if $n$ is odd, replace it by $3n+1$. Repeat this forever. Prove or disprove that every starting value eventually reaches 1.

Let $x_1$ and $x_2$ be the roots of $x^2+px+q=0$. Express $x_1^3+x_2^3$ in terms of $p$ and $q$ without solving the equation.

Twin primes are pairs of primes that differ by 2, such as $11,13$ or $41,43$. Prove that there are infinitely many twin prime pairs.

Prove that every even integer greater than 2 is the sum of two primes. For example $28=5+23$ and $100=3+97$.

Find all polynomials $P$ with real coefficients such that $P(x^2)=P(x{)}^2$ for every real $x$.

Find the value of $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots }}}}$ and explain in what sense the infinite expression has a value.

Prove that for every positive integer $n$ $1^3+2^3+\cdots +n^3=(1+2+\cdots +n{)}^2.$

Color every point of the plane so that any two points at distance exactly 1 receive different colors. What is the smallest number of colors for which this is possible? It is known that the answer is 5, 6 or 7. Determine it.

Prove that for positive real numbers $x_1,\dots ,x_n$ $\frac{x_1+\cdots +x_n}{n}\ge \sqrt[n]{x_1{x}_2\cdots x_n},$ with equality only when all the $x_i$ are equal.

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Prove that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.

Show that there is no formula, built from the coefficients with addition, subtraction, multiplication, division and radicals, that gives the roots of every polynomial equation of degree five.

$n$ runners start together on a circular track of length 1 and run forever, each at its own constant speed, with all speeds different. Prove that for each runner there is a moment when every other runner is at distance at least $1/n$ along the track. The speeds $0,1,\dots ,n-1$ show that $1/n$…

Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that $f(x+y)=f(x)+f(y)$ for all rational $x$ and $y$.

Prove that for all positive real numbers $a$, $b$ and $c$ $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}.$ When does equality hold?

A finite family of finite sets is union-closed if the union of any two members is again a member. Prove that every finite union-closed family that contains at least one nonempty set has an element lying in at least half of the sets of the family.

Solve the system $x+y+z=6,xy+yz+zx=11,xyz=6.$

Let $C$ be any simple closed curve in the plane. Prove or disprove that $C$ always contains four points that are the vertices of a square.

Find a rectangular box whose three edges, three face diagonals and space diagonal all have integer lengths, or prove that none exists. In other words, find positive integers $a,b,c$ such that $a^2+b^2$, $b^2+c^2$, $a^2+c^2$ and $a^2+b^2+c^2$ are all perfect squares.

Prove that for every integer $n\ge 2$ there are positive integers $x,y,z$ with $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

A positive integer is perfect if it equals the sum of its proper divisors, like $6=1+2+3$ and $28=1+2+4+7+14$. Every known perfect number is even. Is there an odd perfect number?