[b]a)[/b] Show that, for any distinct natural numbers $ m,n, $ the rings $ \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 $ are homomorphic, but not isomorphic. [b]b)[/b] Show that there are infinitely many pairwise nonhomomorphic rings of same order.

$1+2+3+4+5+6=6+7+8$. What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?

All diagonals are plotted in a $2017$-sided convex polygon. A line $\ell$ intersects said polygon but does not pass through any of its vertices. Show that the line $\ell$ intersects an even number of diagonals of said polygon.

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.

A factory produces several types of mug, each with two colors, chosen from a set of six. Every color occurs in at least three different types of mug. Show that we can find three mugs which together contain all six colors.

Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$, b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$. Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.

The vertices of a given square are clockwise lettered $A,B,C,D$. On the side $AB$ is situated a point $E$ such that $AE = AB/3$. Starting from an arbitrarily chosen point $P_0$ on segment $AE$ and going clockwise around the perimeter of the square, a series of points $P_0, P_1, P_2, \ldots$ is marked on the perimeter such that $P_iP_{i+1} = AB/3$ for each $i$. It will be clear that when $P_0$ is chosen in $A$ or in $E$, then some $P_i$ will coincide with $P_0$. Does this possibly also happen if $P_0$ is chosen otherwise?

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

We call a set of integers $\textit{special}$ if it has $4$ elements and can be partitioned into $2$ disjoint subsets $\{ a,b \}$ and $\{ c, d \}$ such that $ab - cd = 1$. For every positive integer $n$, prove that the set $\{ 1, 2, \dots, 4n \}$ cannot be partitioned into $n$ disjoint special sets. [i]Proposed by Mohsen Jamali, Iran[/i]

The number of pairs of positive integers $(x,y)$ which satisfy the equation $x^2+y^2=x^3$ is $\textbf {(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf {(C) } 2 \qquad \textbf {(D) } \text{not finite} \qquad \textbf {(E) } \text{none of these}$

For an integer $n \ge 3$ we consider a circle with $n$ points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other. For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

Suppose m and n are positive odd integers. Which of the following must also be an odd integer? $ \textbf{(A)}\ m+3n\qquad\textbf{(B)}\ 3m-n\qquad\textbf{(C)}\ 3m^2 + 3n^2\qquad\textbf{(D)}\ (nm + 3)^2\qquad\textbf{(E)}\ 3mn $

Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent: a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$; b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$

The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A$. The remaining cards form pile $B$. The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A$, respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number $1$ is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named [i]magical[/i]. Find the number of cards in the magical stack in which card number $131$ retains its original position.

A sequence $(a_{n})$ of positive integers is given by $a_{n}=[n+\sqrt{n}+\frac{1}{2}]$. Find all of positive integers which belong to the sequence.

Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.

Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$ for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$. [i]Proposed by Wong Jer Ren[/i]

There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Find all the finite sets $A$ of real positive numbers having at least two elements, with the property that $a^2 + b^2 \in A$ for every $a, b \in A$ with $a \ne b$

There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.