Find the maximum value of the expression $ x+y+z, $ where $ x,y,z $ are real numbers satisfying $$ \left\{ \begin{matrix} x^2+yz\le 2 \\y^2+zx\le 2\\ z^2+xy\le 2 \end{matrix} \right. . $$

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

Three cats--TheInnocentKitten, TheNeutralKitten, and TheGuiltyKitten labelled $P_1, P_2,$ and $P_3$ respectively with $P_{n+3} = P_{n}$--are playing a game with three rounds as follows: [list=1] [*] Each round has three turns. For round $r \in \{1,2,3\}$ and turn $t \in \{1,2,3\}$ in that round, player $P_{t+1-r}$ picks a non-negative integer. The turns in each round occur in increasing order of $t$, and the rounds occur in increasing order of $r$. $\newline \newline$ [*] [b]Motivations:[/b] Every player focuses primarily on maximizing the sum of their own choices and secondarily on minimizing the total of the other players’ sums. TheNeutralKitten and TheGuiltyKitten have the additional tertiary priority of minimizing TheInnocentKitten’s sum. $\newline \newline$ [*] For round $2$, player $P_{2}$ has no choice but to pick the number equal to what player $P_{1}$ chose in round $1$. Likewise, for round $3$, player $P_{3}$ must pick the number equal to what player $P_{2}$ chose in round $2$. $\newline \newline$ [*] If not all three players choose their numbers such that the values they chose in rounds 1,2,3 form an arithmetic progression in that order by the end of the game, all players' sums are set to $-1$ regardless of what they have chosen. $\newline \newline$ [*] If the sum of the choices in any given round is greater than $100$, all choices that round are set to $0$ at the end of that round. That is, rules $2$, $3$, and $4$ act as if each player chose $0$ that round. $\newline \newline$ [*] All players play optimally as per their motivations. Furthermore, all players know that all other players will play optimally (and so on.) [/list] Let $A$ and $B$ be TheInnocentKitten's sum and TheGuiltyKitten's sum respectively. Compute $1000A + B$ when all players play optimally. Proposed by Harry Chen (Extile)

At the math contest each participant met at least $3$ pals who he/she already knew. Prove that the Jury can choose an even number of participants (more than two) and arrange them around a table so that each participant be set between these who he/she knows.

Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Five circles in a row are each labeled with a positive integer. As shown in the diagram, each circle is connected to its adjacent neighbor(s). The integers must be chosen such that the sum of the digits of the neighbor(s) of a given circle is equal to the number labeling that point. In the example, the second number $23 = (1+8)+(5+9)$, but the other four numbers do not have the needed value. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi9lL2M2MzVkMmMyYTRlZjliNWEzYWNkOTM2OGVmY2NkOGZmOWVkN2VmLnBuZw==&rn=MjAxMSBCQU1PIDIucG5n[/img] What is the smallest possible sum of the five numbers? How many possible arrangements of the five numbers have this sum? Justify your answers.

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$

On a table are given $n$ markers, each of which is denoted by an integer. At any time, if some two markers are denoted with the same number, say $k$, we can redenote one of them with $k +1$ and the other one with $k -1$. Prove that after a finite number of moves all the markers will be denoted with different numbers.

In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.

Prove that for arbitrary fixed $a_1, a_2,.. , a_{31}$ the sum $\cos 32x + a_{31} \cos 31x +... + a_2 cos 2x + a_1 \cos x$ can take both positive and negative values as $x$ varies.

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$

Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let $A_1,B_1,C_1$ be the second intersection points of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the intersection point of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the intersection point of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle AMP$ .

Let $k>1$ be a positive integer. On a $\text{k} \times \text{k}$ square grid, Tom and Jerry are on opposite corners, with Tom at the top right corner. Both can move to an adjacent square every move, where two squares are adjacent if they share a side. Tom and Jerry alternate moves, with Jerry going first. Tom [i]catches[/i] Jerry if they are on the same square. We aim to answer to the following question: What is the smallest number of moves that Tom needs to guarantee catching Jerry? (a) Without proof, find the answer in the cases of $k=2,3,4$, and (correctly) guess what the answer is in terms of $k$. We'll refer to this answer as $A(k)$. (b) Find a strategy that Jerry can use to guarantee that Tom takes at least $A(k)$ moves to catch Jerry. Now, you will find a strategy for Tom to catch Jerry in at most $A(k)$ moves, no matter what Jerry does. (c) Find, with proof, a working strategy for $k=5$. (d) Find, with proof, a working strategy for all $k \geq 2$.

The function $f$ is given by the table \[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\] If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

The sum of $n$ consecutive integers is divisible by $n$ for some $n > 1$. For which $n$ is this always true? $\textbf{(A) }\text{even }n\qquad\textbf{(B) }\text{odd }n\text{ divisible by }3\qquad\textbf{(C) }\text{odd }n\qquad\textbf{(D) }\text{prime }n\qquad\textbf{(E) }\text{no such }n\text{ exists}$

In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.

The integers from $1$ to $n$ are arranged along a circle such that each number is a multiple of difference of its adjacents. For which $n$ below such an arrangement is possible? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 13 $

The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle. $\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

[b] Problem Section #3 NOTE: Neglect that HF and CD.