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?

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

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.

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]. $

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 $S_3.$

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]

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?

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.

A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$: $$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$ Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define: $$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$ Investigate a) whether $M$ is a semigroup; b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.