Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.

Find all real polynomials $P(x)$ that satisfy $$P(x^3-2)=P(x)^3-2$$

Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

Let $P(x)$ and $Q(x)$ be non-constant integer-coefficient polynomials such that for any integer $x\in \mathbb Z$, there exists integer $y\in \mathbb Z$ such that $P(x)=Q(y)$. Prove that the degree of $Q$ divides the degree of $P$.

Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

The teacher wrote on the blackboard quadratic polynomial $x^2 + 10x + 20$. Then in turn each student in the class either increased or decreased by $1$ either the coefficient of $x$ or the constant term. At the end the quadratic polynomial became $x^2+20x+10$. Is it true that at certain moment a quadratic polynomial with integer roots was on the board?

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?

Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.

The function $f(x) = ax + b$ satisfies the following equalities: \begin{align*} f(f(f(1))) &= 2023, \\ f(f(f(0))) &= 1996. \end{align*} Find the value of $a$.

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after $2014$ tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of $x^{2013}$ is $2014$, and that the constant term is $1$. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)

There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation \[ \tan(15 x) = 15 \tan(x) . \] Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum \[ \frac{1}{\tan^2 r_1} + \frac{1}{\tan^2 r_2} + \frac{1}{\tan^2 r_3} + \frac{1}{\tan^2 r_4} + \frac{1}{\tan^2 r_5} + \frac{1}{\tan^2 r_6} \, ? \]

Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?

Find all nonzero polynomials $P(x)$ with real coefficents, that satisfies $$P(x)^3+3P(x)^2=P(x^3)-3P(-x)$$ for all real numbers $x$

Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form \[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\] with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.

Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find \[ \left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2. \]

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of $$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$

Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$. [i](Paolo Leonetti)[/i]

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?