Problems

Prove the inequality: $$\frac{(b_1+\dots b_n{)}^{b_1+\dots b_n}}{(a_1+\dots a_n{)}^{b_1+\cdots +b_n}}\le {\left(\frac{b_1}{a_1}\right)}^{b_1}\dots {\left(\frac{b_n}{a_n}\right)}^{b_n}$$ where all variables are considered positive.

Prove that the polynomial $P(x)$ is divisible by its derivative if and only if $P(x)$ has the form $P(x)=a_n(x-x_0{)}^n$.

A polynomial of degree $n>1$ has $n$ distinct roots $x_1,x_2,\dots ,x_n$. Its derivative has the roots $y_1,y_2,\dots ,y_{n-1}$. Prove the inequality $$\frac{x_1^2+\cdots +x_n^2}{n}>\frac{y_1^2+\cdots +y_n^2}{n}.$$

Solve the equation $2\sin \pi x/2-2\cos \pi x=x^5+10x-54$.

Solve the equation $f(f(x))=f(x)$ if $f(x)=\sqrt[5]{3-x^3-x}$.

George drew an empty table of size $50\times 50$ and wrote on top of each column and to the left of each row, a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down the sum of the numbers written at the start of the corresponding row and column (“addition table”). What is the largest number of sums in this table that could be rational numbers?

The number $n$ has the property that when it is divided by $q^2$ the remainder is smaller than $q^2/2$, whatever the value of $q$. List all numbers that have this property.