Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$. a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points. b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.

Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$). It is assumed that no-one can have infinitely many children. [i]Variant 1[/i]. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$). [i]Variant 2.[/i] Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$.

One quadrant of the Cartesian coordinate system is tiled with $1\times 2$ dominoes. The dominoes don’t overlap with each other, they cover the entire quadrant and they all fit in the quadrant. Farringdon the flea is initially sitting at the origin and is allowed to jump from one corner of a domino to the opposite corner any number of times. Is it possible that the dominoes are arranged in such a way that Farringdon is unable to move to a distance greater than 2023 from the origin?

In a country, some cities are connected by two-way airlines, and one can get from any city to any other city in no more than $n{}$ flights. Prove that all airlines can be distributed among $n{}$ companies so that a route can be built between any two cities in which no more than two flights of each company would meet. [i]From the folklore[/i]

155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.

Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.

On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?

In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly $100$ roads. We call $10$ roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.

In a country, there are ${}N{}$ cities and $N(N-1)$ one-way roads: one road from $X{}$ to $Y{}$ for each ordered pair of cities $X \neq Y$. Every road has a maintenance cost. For each $k = 1,\ldots, N$ let's consider all the ways to select $k{}$ cities and $N - k{}$ roads so that from each city it is possible to get to some selected city, using only selected roads. We call such a system of cities and roads with the lowest total maintenance cost $k{}$[i]-optimal[/i]. Prove that cities can be numbered from $1{}$ to $N{}$ so that for each $k = 1,\ldots, N$ there is a $k{}$-optimal system of roads with the selected cities numbered $1,\ldots, k$. [i]Proposed by V. Buslov[/i]

Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?

Let \(L_3 = \{3\}\), \(L_n = \{3, 4, \ldots, h\}\) (where \(h > 3\)). For any given integer \(n \geq 3\), consider a graph \(G\) with \(n\) vertices that contains a Hamiltonian cycle \(C\) and has more than \(\frac{n^2}{4}\) edges. For which lengths \(l \in L_n\) must the graph \(G\) necessarily contain a cycle of length \(l\)?

Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.

Graph $G$ has $n$ vertices and $mn$ edges, where $n>2m$, show that there exists a path with $m+1$ vertices. (A path is an open walk without repeating vertices )

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

A school has 2021 students, each of which knows at least 45 of the other students (where "knowing" is mutual). Show that there are four students who can be seated at a round table such that each of them knows both of her neighbours.

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) If $G_{n}$ does not contain $K_{2,3}$ , prove that $e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n$. (b) Given $n \geq 16$ distinct points $P_{1}, . . . , P_{n}$ in the plane, prove that at most $n\sqrt{n}$ of the segments $P_{i}P_{j}$ have unit length.

In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be?

For $n$ people, if it is known that (a) there exist two people knowing each other among any three people, and (b) there exist two people not knowing each other among any four people. Find the maximum of $n$. Here, we assume that if $A$ knows $B$, then $B$ knows $A$.

Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion: For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.