#algebra

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.

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 $n$ $1^3+2^3+\cdots +n^3=(1+2+\cdots +n{)}^2.$

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.

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?

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

Let $\vee$ be a commutative and associative binary operation and $\neg$ a unary operation on a set, satisfying Robbins's equation $\neg (\neg (a\vee b)\vee \neg (a\vee \neg b))=a$ for all $a,b$. Prove that the structure is a Boolean algebra, or find a counterexample.

Using only a compass and an unmarked straightedge, can one construct in finitely many steps a square with exactly the same area as a given circle? Prove your answer.

A regular tetrahedron and a cube have the same volume. Can the tetrahedron be cut into finitely many polyhedral pieces that can be reassembled to form the cube? Prove your answer.