Find all pairs $ (m,n)$ of integer numbers $ m,n > 1$ with property that $ mn \minus{} 1\mid n^3 \minus{} 1$.

Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.

A rectangle has area $A \text{ cm}^2$ and perimeter $P \text{ cm}$, where $A$ and $P$ are positive integers. Which of the following numbers cannot equal $A+P$? $ \textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108 $

Iris does not know what to do with her 1-kilogram pie, so she decides to share it with her friend Rosabel. Starting with Iris, they take turns to give exactly half of total amount of pie (by mass) they possess to the other person. Since both of them prefer to have as few number of pieces of pie as possible, they use the following strategy: During each person's turn, she orders the pieces of pie that she has in a line from left to right in increasing order by mass, and starts giving the pieces of pie to the other person beginning from the left. If she encounters a piece that exceeds the remaining mass to give, she cuts it up into two pieces with her sword and gives the appropriately sized piece to the other person. When the pie has been cut into a total of 2017 pieces, the largest piece that Iris has is $\frac{m}{n}$ kilograms, and the largest piece that Rosabel has is $\frac{p}{q}$ kilograms, where $m,n,p,q$ are positive integers satisfying $\gcd(m,n)=\gcd(p,q)=1$. Compute the remainder when $m+n+p+q$ is divided by 2017. [i]Proposed by Yannick Yao[/i]

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

In a very large City, they are building a subway: there are many stations, some pairs of which are connected by tunnels, and from any station you can get through tunnels to any other. All metro tunnels must be divided into "lines": each line consists of several consecutive tunnels, all stations in which are different (in particular, the line should not be circular); lines consisting of one tunnel are also allowed. By law, it is required that you can get from any station to any other station by making no more than $100$ transfers from line to line. At what is the largest $N$, any connected metro with $N$ stations can be divided into lines, observing the law?

We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?

Consider a $10 \times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10\%$ of the burrito's original size and accidentally throw it into a random square (possibly the one it is already in). Otherwise, the pigeon will move to an adjacent square, decreasing the distance between it and the burrito. What is the expected number of minutes before the pigeon has eaten the entire burrito?

In rectangle $ABCD$, $AB = 7$ and $AC = 25$. What is its area?

Given is a natural number $n > 5$. On a circular strip of paper is written a sequence of zeros and ones. For each sequence $w$ of $n$ zeros and ones we count the number of ways to cut out a fragment from the strip on which is written $w$. It turned out that the largest number $M$ is achieved for the sequence $11 00...0$ ($n-2$ zeros) and the smallest - for the sequence $00...011$ ($n-2$ zeros). Prove that there is another sequence of $n$ zeros and ones that occurs exactly $M$ times.

Given is that $x, y, z$ are real numbers, different from 0, $x$ and $z$ are different, such that $(x+y) ^2+(2-xy)=9$ and $(y+z) ^2-(3+yz)=4$ Find the value of $A=(x/y+y^2/x^2+z^3/x^2y)(y/z+z^2/y^2+x^3/y^2z)(z/x+x^2/z^2+y^3/z^2x)=?$

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?

Quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle. Altitude $DE$ in triangle $ABD$ intersects diagonal $AC$ in $F$. Prove that $FB = BC$

The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.

Let $n$ be an positive integer. We call $n$ $\textit{good}$ when there exists positive integer $k$ s.t. $n=k(k+1)$. Does there exist 2022 pairwise distinct $\textit{good}$ numbers s.t. their sum is also $\textit{good}$ number?

Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. [img]https://i.imgur.com/1LU5Zmb.png[/img]

Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$. Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. [i]Proposed by Ariel García[/i]

Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Lewis Chen [/i]

In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And (i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$. (ii) $a \circ b \neq b \circ a$ when $a \neq b$. Prove that: a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$. b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.

An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly 8 moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? [asy] unitsize(0.25cm); pair A, B, C, O; A = (-8, 0); B = (8, 0); C = (0, 15); O = (0, 0); draw(arc(O, 120/17, 0, 180)); draw(A--B--C--cycle); [/asy] $\textbf{(A) }4 \sqrt{3}\qquad\textbf{(B) } \dfrac{120}{17}\qquad\textbf{(C) }10\qquad\textbf{(D) }\dfrac{17\sqrt{2}}{2}\qquad \textbf{(E) }\dfrac{17\sqrt{3}}{2}$

If a b c positive reals smaller than 1, prove: a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)