$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

Find all functions $f: \mathbb{R^{+}} \rightarrow \mathbb {R^{+}}$ such that $f(f(x)+\frac{y+1}{f(y)})=\frac{1}{f(y)}+x+1$ for all $x, y>0$. [i]Proposed by Dominik Burek, Poland[/i]

In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$, $d > 0$. \[ \begin{array}{l} a_{11} \,\, a_{12} \,\, ... \,\, a_{1n}\\ a_{21} \,\, a_{22} \,\, ... \,\, a_{2n}\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ a_{n1} \,\, a_{n2} \,\, ... \,\, a_{nn}\\ \end{array} \] Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.

For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?

Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by: $ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$ contains infinitely many natural numbers.

Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

Positive numbers $x$ and $y$ satisfy $xy=2^{15}$ and $\log_2{x} \cdot \log_2{y} = 60$. Find $\sqrt[3]{(\log_2{x})^3+(\log_2{y})^3}$

Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum: $$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$ [i]Proposed by Lia Chitishvili, Georgia [/i]

Find the smallest natural number $n$ for which the equality $\sin n^o= \sin (1997n)^o$ holds.

Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.

Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$

Let $r>s$ be positive integers. Let $P(x)$ and $Q(x)$ be distinct polynomials with real coefficients, non-constant(s), such that $P(x)^r-P(x)^s=Q(x)^r-Q(x)^s$ for every $x\in \mathbb{R}$. Prove that $(r,s)=(2,1)$.

Two perpendicular lines pass through the point $F(1;1)$ of coordinate plane. One of them intersects hyperbola $y=\frac{1}{2x}$ at $A$ and $C$ ($C_x>A_x$), and the other one intersects the left part of hyperbola at $B$ and the right at $D$. Let $m=(C_x-A_x)(D_x-B_x)$ Find the area of non-convex quadraliteral $ABCD$ (in terms of $m$)

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $ \forall x\notin\{-1,1\}$ holds: \[\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}\]

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$.

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]