Knowing that the polynomials $$2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25$$ $$x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50$$ have two common double roots, determine all their roots.

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.

The roots of the equation $x^5-180x^4+Ax^3+Bx^2+Cx+D=0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality: $$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$ (Karl Czakler)

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

You are standing at a pole and a snail is moving directly away from the pole at $1$ cm/s. When the snail is $1$ meter away, you start "Round 1". In Round $n$ ($n\ge 1$), you move directly toward the snail at $n+1$ cm/s. When you reach the snail, you immediately turn around and move back to the starting pole at $n + 1$ cm/s. When you reach the pole, you immediately turn around and Round $n + 1$ begins. At the start of Round $100$, how many meters away is the snail?

Let $a_1,a_2,a_3,\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n.$ Suppose $a_1 = 1,$ $a_2 = 3,$ \[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] Then $c - b + \tfrac{c-b}{b}$ can be written in the form $x^y,$ where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y.$

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

A sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $0 \leq a_1 \leq 1$ and $a_{n+1} = \tfrac{1 + \sqrt{a_n}}{2}$ for all positive integers $n$. If $a_1 + a_{2021} = 1$, then the product $a_1a_2a_3\cdots a_{2020}$ can be written in the form $m^k$, where $k$ is an integer and $m$ is a positive integer that is not divisible by any perfect square greater than $1$. Compute $m + k$.

Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds: $$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$

Given are real numbers $a_1, a_2, \ldots, a_n$ ($n>3$), such that $a_k^3=a_{k+1}^2+a_{k+2}^2+a_{k+3}^2$ for all $k=1,2,...,n$. Prove that all numbers are equal.

Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.

Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.

It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions: (a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and (b) $\lim_{x\to \infty} f(x) = 0$.

Art and Kimberly build flagpoles on a level ground with respective heights $10$ m and $15$ m, separated by a distance of $5$ m. Kimberly wants to move her flagpole closer to Art’s, but she can only doing so in the following manner: 1. Run a straight wire from the top of her flagpole to the bottom of Art’s. 2. Run a straight wire from the top of Art’s flagpole to the bottom of hers. 3. Build the flagpole to the point where the wires meet. If Kimberly keeps moving her flagpole in this way, compute the number of flagpoles she will build whose heights are $1$ m or greater (not counting her original $15$ m flagpole).

For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$. (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions. [list=disc] [*] $f(a)$ is not an integer for some rational number $a$. [*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers. [/list]

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]