Problems

In what number system is the equality $3\times 4=10$ correct?

Prove that $\frac{1}{2}(x^2+y^2)\ge xy$ for any $x$ and $y$.

Prove that for $a,b,c>0$, the following inequality is valid: ${\left(\frac{a+b+c}{3}\right)}^2\ge \frac{ab+bc+ca}{3}$.

Prove that for $x\ge 0$ the inequality is valid: $2x+\frac{3}{8}\ge \sqrt[4]{x}$.

Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?

Prove that any axis of symmetry of a 45-gon passes through its vertex.

Is the number $1+2+3+\cdots +1990$ odd or even?

Every Martian has three hands. Can seven Martians join hands?

At the vertices of a $n$-gon are the numbers $1$ and $-1$. On each side is written the product of the numbers at its ends. It turns out that the sum of the numbers on the sides is zero. Prove that a) $n$ is even; b) $n$ is divisible by 4.

There are 30 people, among which some are friends. Prove that the number of people who have an odd number of friends is even.