Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.

For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that $\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$ $\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that \[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.

The functions $f,g:[0,1]\to\mathbb R$ are given with the formulas $$f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),$$ and $c$ denotes any solution of $x=f(x)$. (a) i. Analyze the function $f(x)$ and draw its graph. Prove that the equation $f(x)=x$ has the unique root $c$ satisfying $c\in[0.72,0.73]$. ii. Analyze the function $f'(x)$. Let $M_1$ and $M_2$ be the points of the graph of $f(x)$ with different $x$ coordinates. What is the position of the arc $M_1M_2$ of the graph with respect to the segment $M_1M_2$? iii. Analyze the function $g(x)$ and draw its graph. What is the position of that graph with respect to the line $y=x$? Find the tangents to the graph at points with $x$ coordinates $0$ and $1$. iv. Prove that every sequence $\{a_n\}$ with the conditions $a_1\in(0,1)$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$ converges. [hide=Official Hint]Consider the sequences $\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)$ and the function $g(x)$ associated with the graph.[/hide] (b) On the graph of the function $f(x)$ consider the points $M$ and $M'$ with $x$ coordinates $x$ and $f(x)$, where $x\ne c$. i. Prove that the line $MM'$ intersects with the line $y=x$ at the point with $x$ coordinate $$h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.$$ ii. Prove that if $x\in(0,c)$ then $h(x)\in(x,c)$. iii. Analyze whether the sequence $\{a_n\}$ satisfying $a_1\in(0,c),a_{n+1}=h(a_n)$ for $n\in\mathbb N$ converges. Prove that the sequence $\{\tfrac{a_{n+1}-c}{a_n-c}\}$ converges and find its limit. (c) Assume that the calculator approximates every number $b\in[-2,2]$ by number $\overline b$ having $p$ decimal digits after the decimal point. We are performing the following sequence of operations on that calculator: 1) Set $a=0.72$; 2) Calculate $\delta(a)=\overline{f(a)}-a$; 3) If $|\delta(a)|>0.5\cdot10^{-p}$, then calculate $\overline{h(a)}$ and go to the operation $2)$ using $\overline{h(a)}$ instead of $a$; 4) If $|\delta(a)|\le0.5\cdot10^{-p}$, finish the calculation. Let $\overleftrightarrow c$ be the last of calculated values for $\overline{h(a)}$. Assuming that for each $x\in[0.72,0.73]$ we have $\left|\overline{f(x)}-f(x)\right|<\epsilon$, determine $\delta(\overleftrightarrow c)$, the accuracy (depending on $\epsilon$) of the approximation of $c$ with $\overleftrightarrow c$. (d) Assume that the sequence $\{a_n\}$ satisfies $a_1=0.72$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$. Find the smallest $n_0\in\mathbb N$, such that for every $n\ge n_0$ we have $|a_n-c|<10^{-6}$.

Given a function $f$ for which \[f(x)=f(398-x)=f(2158-x)=f(3214-x) \]holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\ldots,f(999)?$

Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane. 30 points

For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that \[ W(x) e^{W(x)} = x. \] The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that \[ \frac{y}{\log_{2} y} = - \frac{3}{5} \, . \] The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

Let $f : \mathbb N \to \mathbb N$ be a function satisfying \[f(f(m)+f(n))=m+n, \quad \forall m,n \in \mathbb N.\] Prove that $f(x)=x$ for all $x \in \mathbb N$.

[list=a] [*] Let $a<b$ and $f:[a,b]\rightarrow\mathbb{R}$ be a strictly monotonous function such that $\int_a^b f(x) dx=0$. Show that $f(a)\cdot f(b)<0$. [*] Find all convergent sequences $(a_n)_{n\geq 1}$ for which there exists a scrictly monotonous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{a_{n-1}}^{a_n} f(x)dx = \int_{a_n}^{a_{n+1}} f(x)dx,\text{ for all }n\geq 2.$$

Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa) (proposed by B. Batbaysgalan, folklore).

Let $a,b,c$ be positive real numbers with $a+b>c$. Prove that $ax + \sin(bx) + \cos(cx) > 1$ for all $x\in \left(0, \frac{\pi}{a+b+c}\right)$.

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$ for all positive $x,y$, then $$2f(\sqrt{xy})=f(x)+f(y)$$ for every pair $x,y$ of positive numbers.

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.

a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$

For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of $$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$