Problems

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.

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 $$\frac{x_1^2+\cdots +x_n^2}{n}>\frac{y_1^2+\cdots +y_n^2}{n}.$$

Hercules meets the three-headed snake Hydra of Lerna. Every minute, Hercules chops off one head of the snake. Let $x$ be the survivability of the snake ($x>0$). The probability $p_s$ of the fact that in the place of the severed head will grow s new heads $(s=0,1,2)$ is equal to $\frac{x^s}{1+x+x^2}$.

During the first 10 minutes of the battle, Hercules recorded how many heads grew in place of each chopped off one. The following vector was obtained: $K=(1,2,2,1,0,2,1,0,1,2)$. Find the value of the survivability of the snake, under which the probability of the vector $K$ is greatest.

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.

Find the minimum for all $\alpha$, $\beta$ of the maximum of the function $y(x)=|\cos x+\alpha \cos 2x+\beta \cos 3x|$.