a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$

Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$. [i]Proposed by P. Volkmann[/i]

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

Call a number ``Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$.

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

In a country, one can get from some point $A$ to any other point either by walking, or by calling a cab, waiting for it, and then being driven. Every citizen always chooses the method of transportation that requires the least time. It turns out that the distances and the traveling times are as follows: $1$ km takes $10$ min, $2$ km takes $15$ min, $3$ km takes $17.5 $ min. We assume that the speeds of the pedestrian and the cab, and the time spent waiting for cabs, are all constants. How long does it take to reach a point which is $6$ km from $A$?

Find the maximum possible value of the product of distinct positive integers whose sum is $2004$.

For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and \[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\ 2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases} \] Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.

In one night the air temperature remained constant, several degrees below zero, and that of the water of a very extensive cylindrical pond, which formed a layer $10$ cm deep, it reached zero degrees, beginning then to form a layer of ice on the surface. Under these conditions it can be assumed that the thickness of the ice sheet formed is directly proportional to the square root of the time elapsed. At $0$ h, the thickness of the ice was $3$ cm and at $4$ h it was just over to freeze the water in the pond. Calculate at what time the ice sheet began to form, knowing that the density of the ice formed was $0.9$.

You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is $.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a decimal.

For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$

Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$. [i]Proposed by Zachary Perry[/i]

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Show that: [b]a)[/b] $ f $ is nondecreasing, if $ f+g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]b)[/b] $ f $ is nondecreasing, if $ f\cdot g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [i]Cristian Mangra[/i]

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$. a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic. b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.

We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$. [i]Solution by Mostafa Eynollahzade and Erfan Salavati[/i]

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

Given $2017$ positive real numbers $a_1,a_2,\dots ,a_{2017}$. For each $n>2017$, set $$a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.$$ Prove that there exists a positive integer $m\leq 2017$ and a positive integer $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for every $n>N$.

Compute the positive integer $n$ such that $\log_3 n < \log_2 3 < \log_3 (n + 1)$.

Let $a$ and $b$ be positive integers satisfying \[ \frac a{a-2} = \frac{b+2021}{b+2008} \] Find the maximum value $\dfrac ab$ can attain.

Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$ for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$. [i]Proposed by Navilarekallu Tejaswi[/i]

There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$.