Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$. \[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]

The function $f$ is given by the table \[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\] If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions. (i) $\{ f(1),f(2),\dots,f(n)\} = S$ (ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.

Prove that these three statements are equivalent: (a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$. (b) There is no antipodal mapping $f:S^n \to S^{n-1}$. (c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.

Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

Let $ I $ be a real interval, and $ f:I\longrightarrow\mathbb{R} $ be a continuous function. Prove that $ f $ is monotone if and only if $ \min(\left( f(a),f(b)\right) \le\frac{1}{b-a}\int_a^b f(x)dx \le\max\left( f(a),f(b) \right) , $ for any distinct $ a,b\in I. $

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

Define the function $f_k(x)$ (where $k$ is a positive integer) as follows: \[f_k(x)=(\cos kx)(\cos x)^k+(\sin kx)(\sin x)^k-(\cos 2x)^k.\] Find the sum of all distinct value(s) of $k$ such that $f_k(x)$ is a constant function.

Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ \plus{} }$: \[ x\sqrt y \plus{} y\sqrt z \plus{} z\sqrt x\leq K\sqrt {(x \plus{} y)(y \plus{} z)(z \plus{} x)} \]

Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that \[ \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$. Prove that $ f''(c)\equal{}0$.

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

Let us call a continuous function $ f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $ a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $ h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $ f(t)\equal{}f(h(t))$ is satisfied for every $ \alpha \leq t \leq \beta$); otherwise $ f$ is irreducible. Construct irreducible $ f : [a,b] \rightarrow \mathbb{R}^2$ and $ g : [c,d] \rightarrow \mathbb{R}^2$ such that $ f([a,b])\equal{}g([c,d])$ and (a) both $ f$ and $ g$ are rectifiable but their lengths are different; (b) $ f$ is rectifiable but $ g$ is not. [i]A. Csaszar[/i]

Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$, $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$. Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.

Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.

$a_1, a_2, ..., a_{95}$ are positive reals. Show that $\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$

Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.

Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.

For a positive integer $n$ with prime factorization $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ let's define $\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}$. Define $L(n)$ as sum of $\lambda(x)$ over all integers from $1$ to $n$. Define $K(n)$ as sum of $\lambda(x)$ over all [b]composite[/b] integers from $1$ to $n$. For some $N>1$, we know, that for every $2\le n \le N$, $L(n)\le 0$. Prove that for this $N$, for every $2\le n \le N$, $K(n)\ge 0$. [i]Mykhailo Shtandenko[/i]

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$, a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m\minus{}1$ players in $ S$ and one player who lost all her games against the other $ m \minus{} 1$ players in $ S$. For a given integer $ m \ge 4$, determine the minimum value of $ n$ (as a function of $ m$) such that the following holds: in every $ n$-player round robin chess tournament with property $ P(m)$, the final scores of the $ n$ players are all distinct.

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them. Find maximum of $\mid S\mid$. [i]Proposed by Erfan Salavati[/i]