Find all triples of integers \( (x, y, z) \) such that \[ x - yz = 11 \] \[ xz + y = 13 \]

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties: i) $P(0) = 1$ ii) for any real numbers $x$ and $y,$ \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]

Prove that for arbitrary real numbers $a$ and $b$ the following inequality is true $$a^2 +ab+b^2 \geq 3(a+b-1).$$

Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$ [asy] unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(",",(20,0)); label(",",(29,0)); label(",...",(35.5,0)); [/asy] Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth? $\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)$

The five solutions to the equation $$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Recall that the [i]eccentricity[/i] of an ellipse $\mathcal E$ is the ratio $\frac ca$, where $2a$ is the length of the major axis of $E$ and $2c$ is the is the distence between its two foci.) $\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15$ Proposed by [b]djmathman[/b]

A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$, $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$. Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$

If, in the expression $ x^2 \minus{} 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount: $ \textbf{(A)}\ {\pm 2ax \plus{} a^2}\qquad \textbf{(B)}\ {2ax \pm a^2}\qquad \textbf{(C)}\ {\pm a^2 \minus{} 3} \qquad \textbf{(D)}\ {(x \plus{} a)^2 \minus{} 3}\qquad\\ \textbf{(E)}\ {(x \minus{} a)^2 \minus{} 3}$

On a $2\times 40$ chessboard colored black and white in the standard alternating pattern, $20$ rooks are placed randomly on the black squares. The expected number of white squares with only rooks as neighbors can be expressed as $a/b$, where $a$ and $b$ are coprime positive integers. What is $a + b$? (Two squares are said to be neighbors if they share an edge.)

Determine the greatest integer $N$ such that $N$ is a divisor of $n^{13}-n$ for all integers $n$.

Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?

There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/i] to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

We are writing either $0$ or $1$ to unit squares of an $8\times 8 $ chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of the all numbers on the board? $ \textbf{(A)}\ 32 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 52 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 64 $

Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.