$n$ points are given on a plane where $n\ge4$. All pairs of points are connected with a segment. Find the maximal number of segments which don't intersect with any other segments in their interior.

Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, $$f(x+L)=f(x),\enspace g(x+L)=g(x).$$

Two friends walked towards each other along a straight road. Each had a constant speed but one was faster than the other. At one moment each friend released his dog to run freely forward, the speed of each dog is the same and constant. Each dog reached the other person and then returned to its owner. Which dog returned to its owner the first, of the person who walks fast or who walks slow?

Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.

A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least $5$?

If $ 8\cdot2^x \equal{} 5^{y \plus{} 8}$, then when $ y \equal{} \minus{} 8,x \equal{}$ $ \textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ \minus{} 3 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8$

In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

For a positive integer $n$, we say an $n$-[i]transposition[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$. Fix some four pairwise distinct $n$-transpositions $\sigma_1,\sigma_2,\sigma_3,\sigma_4$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$, \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\] (Note that the equalities in the previous sentence are in $\mathbb F_q$. Note that, for any $a_1,\ldots ,a_n, b_1, \ldots , b_n \in \mathbb{F}_q$, we have $(a_1,\ldots , a_n)+(b_1, \ldots, b_n)=(a_1+b_1,\ldots, a_n+b_n)$, where $a_1+b_1,\ldots , a_n+b_n \in \mathbb{F}_q$.) For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions. Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3,\sigma_4)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)-1}{q-1}\right)\right).\] Pick four pairwise distinct $n$-transpositions $\sigma_1,\sigma_2,\sigma_3,\sigma_4$ uniformly at random from the set of all $n$-transpositions. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\ldots,\sigma_4)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$. [i]Proposed by Gopal Goel[/i]

Let circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. Let the line externally tangent to both circles that is closer to $Q$ touch $\omega_1$ at $A$ and $\omega_2$ at $B$. Let point $T$ lie on segment$ P Q$ such that $\angle AT B = 90^o$. Given that $AT = 6$, $BT = 8$, and $P T = 4$, compute $P Q$.

What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order?

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$, \[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\] ($\mathbb{R}^+$ denotes the set of positive real numbers.) [i](Proposed by Ivan Chan Guan Yu)[/i]

There are $n$ points on a circle, such that each line segment connecting two points is either red or blue. $P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$. (a) For which values of $n$ is this possible? (b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.

If $x_k \in \mathbb{R}$ ($k=1, 2, \cdots , n$) and $m \in \mathbb{N}$ then [list=1] [*] $\sum \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq 3^m \sum \limits_{k=1}^n x_k^{2m}$ [*] $\prod \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq \left (\frac{3^m}{n}\right )^m \left (\sum \limits_{k=1}^n x_k^{2m}\right )^n$ [/list]

Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ in which $AB + CD = AD$. Diagonals $AC$ and $BD$ intersect in point $E$. Line passing through point $E$ and parallel to bases of trapezium cuts $AD$ in point $F$. Prove that $\sphericalangle BFC = 90 ^{\circ}$.

For a positive integer k, let $f_{1}(k)$ be the square of the sum of the digits of k. (For example $f_{1}(123)=(1+2+3)^{2}=36$.) Let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of the $f_{2007}(2^{2006})$. Justify your claim.

Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$). 1) If $k=2n+1$, prove that there exists a person who knows all others. 2) If $k=2n+2$, give an example of such a group in which no-one knows all others.

$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.

Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe?

Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Let point $C$ be such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ at points $X$ and $Y$, respectively. Prove that $CX = CY$. [i]Proposed by Oleksii Masalitin[/i]

Jacek has $n$ cards numbered consecutively with the numbers $1,. . . , n$, which he places in a row on the table, in any order he chooses. Jacek will remove cards from the table in the sequence consistent with the numbering of cards: first they will remove the card number $1$, then the card number $2$, and so on. Before Jacek starts taking the cards, Pie will color each one of cards in red, blue or yellow. Prove that Pie can color the cards in such a way that when Jacek takes them off, it will be fulfilled at every moment the following condition: between any two cards of the same suit there is at least one card of a different color.

Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An [i]eligible[/i] set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. [i]Proposed by Cristi Săvescu[/i]