Olympiad problems

2021 Durer Math Competition Finals, 6

Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.

1969 Vietnam National Olympiad, 1

Tags: combinatorics , graph , subset

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.

2016 IFYM, Sozopol, 1

Tags: combinatorics , graph

There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

2023 Mongolian Mathematical Olympiad, 3

Tags: combinatorics , extremal , graph

Five girls and five boys took part in a competition. Suppose that we can number the boys and girls $1, 2, 3, 4, 5$ such that for each $1 \leq i,j \leq 5$, there are exactly $|i-j|$ contestants that the girl numbered $i$ and the boy numbered $j$ both know. Let $a_i$ and $b_i$ be the number of contestants that the girl numbered $i$ knows and the number of contestants that the boy numbered $i$ knows respectively. Find the minimum value of $\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)$. (Note that for a pair of contestants $A$ and $B$, $A$ knowing $B$ doesn't mean that $B$ knows $A$ and a contestant cannot know themself.)