Let $S$ be the set of values assumed by the fraction \[\frac{2x+3}{x+2}\] when $x$ is any member of the interval $x \ge 0$. If there exists a number $M$ such that no number of the set $S$ is greater than $M$, then $M$ is an upper bound of $S$. If there exists a number $m$ such that such that no number of the set $S$ is less than $m$, then $m$ is a lower bound of $S$. We may then say: $ \textbf{(A)}\ \text{m is in S, but M is not in S} $ $\textbf{(B)}\ \text{M is in S, but m is not in S}$ $\textbf{(C)}\ \text{Both m and M are in S} $ $\textbf{(D)}\ \text{Neither m nor M are in S}$ $\textbf{(E)}\ \text{M does not exist either in or outside S} $

Consider the set $A=\{1,2,3,4,5,6,7,8,9\}$.A partition $\Pi $ of $A$ is collection of disjoint sets whose union is $A$. For example, $\Pi_1=\{\{1,2\},\{3,4,5\},\{6,7,8,9\}\}$ and $\Pi _2 =\{\{1\},\{2,5\},\{3,7\},\{4,5,6,7,8,9\}\}$ can be considered as partitions of $A$. For, each $\Pi$ partition ,we consider the function $\pi$ defined on the elements of$A$. $\pi (x)$ denotes the cardinality of the subset in $\Pi$ which contains $x$. For, example in case of $\Pi_1$ , $\pi_1(1)=\pi_1(2)=2$, $\pi_1(3)=\pi_1(4)=\pi_1 (5)=3$, and $\pi_1(6)=\pi_1(7)=\pi_1(8)=\pi_1(9)=4$. For $\Pi_2$ we have $\pi_2(1)=1$, $\pi_2(2)=\pi_2(5)=2$, $\pi_2(3)=\pi_2(7)=2$ and $\pi_2(4)=\pi_2(6)=\pi_2(8)=\pi_2(9)=4$ Given any two partitions $\Pi$ and $\Pi '$, show that there are two numbers $x$ and $y$ in $A$, such that $\pi (x)= \pi '(x)$ and $ \pi (y)= \pi'(y)$.[[b]Hint:[/b] Consider the case where there is a block of size greater than or equal to 4 in a partition and the alternative case]

There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury $a)$ Prove that we can select $7$ jury such that any student likes at least one jury. $b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.

In a country there are $n$ major cities, $n \ge 4$, connected by railroads, such that each city is directly connected to each other city. Each railroad company in that country has but only one train, which connects a series of cities, at least two, such that the train doesn’t pass through the same city twice in one shift. The companies divided the market such that any two cities are directly$^1$ connected only by one company. Prove that among any $n+1$ companies, there are two which have no common train station or there are two cities that are connected by two trains belonging to two of these $n+1$ companies. $^1$ directly connected means that they are connected by a railroad, without no other station between them

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $S_n=\sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}$. What is the value of $\sum_{n=1}^{99}\frac{1}{S_n+S_{n-1}}$ ?

Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties: (a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$. (b) We have $f\left(2\right) = 0$. (c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$. [b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.

Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?

Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. $k$ can be at most ? $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$

The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations? [i]Proposed by Michael Kural[/i]

Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.

Two reals $x$ and $y$ are such that $x-y=4$ and $x^3-y^3=28$. Compute $xy$.

Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$. (Karl Czakler)

On the base $AB$ of the isosceles triangle $ABC$, lies the point $P$ such that $AP : PB = 1 : 2$. Determine the minimum of $\angle ACP$.

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

Let $\omega_1$ and $\omega_2$ be two circles that intersect at point $A$ and $B$. Define point $X$ on $\omega_1$ and point $Y$ on $\omega_2$ such that the line $XY$ is tangent to both circles and is closer to $B$. Define points $C$ and $D$ the reflection of $B$ WRT $X$ and $Y$ respectively. Prove that the angle $\angle{CAD}$ is less than $90^{\circ}$

Inside an equilateral triangle of side length $6$, three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/3/f/ff48c885154ce065c0d0420d1580769aa98eb1.png[/img] A smaller equilateral triangle formed between the three congruent equilateral triangles has side length $1$. Compute $x$.

Let $P$ be a point inside an acute triangle $ABC$ such that $\angle BPC=90^\circ$. We build triangles $AQB$ and $ARC$, outside of the triangle $ABC$, such that $\angle ABQ = \angle PBC$, $\angle QAB = \angle PAC$, $\angle RCA = \angle PCB$, and $\angle CAR = \angle BAP$. Prove that $P$, $Q$, $R$ are collinear.

Numbers $p,q,r$ lies in the interval $(\frac{2}{5},\frac{5}{2})$ nad satisfy $pqr=1$. Prove that there exist two triangles of the same area, one with the sides $a,b,c$ and the other with the sides $pa,qb,rc$.

Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.

Does there exist an irrational number $\alpha > 1$ such that \[\lfloor \alpha^n \rfloor \equiv 0 \pmod{2017}\] for all integers $n \ge 1$?

A point $C$ lies on the bisector of an acute angle with vertex $S$. Let $P$, $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$. Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent. Proposed by: A.Zaslavsky

$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.

Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?