Olympiad problems

1984 Brazil National Olympiad, 2

Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

1996 Romania National Olympiad, 1

Tags: superior algebra , group theory , Groups , group isomorphism

Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$

2015 BAMO, 2

Tags: counting , Groups , B8 , combinatorics , Olympiad , committees

Members of a parliament participate in various committees. Each committee consists of at least $2$ people, and it is known that every two committees have at least one member in common. Prove that it is possible to give each member a colored hat (hats are available in black, white or red) so that every committee contains at least $2$ members with different hat colors.

2019 District Olympiad, 1

Tags: endomorphism , abtract algebra , Groups , superior algebra

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

2000 District Olympiad (Hunedoara), 1

Tags: Binary operation , semigroups , Groups , superior algebra , group theory

Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

ICMC 5, 4

Tags: reflection , Groups , number theory , mod p , college contests , ICMC

Let $p$ be a prime number. Find all subsets $S\subseteq\mathbb Z/p\mathbb Z$ such that 1. if $a,b\in S$, then $ab\in S$, and 2. there exists an $r\in S$ such that for all $a\in S$, we have $r-a\in S\cup\{0\}$. [i]Proposed by Harun Khan[/i]

2018 District Olympiad, 2

Tags: monoid , Groups

Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^p \ne a$, for any $a\in M \backslash \{e\}$, where $e$ is the identity element of $M$. Show that $(M, \cdot)$ is a group.