The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.

Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.

Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\ \textbf{(V) }2007 & &\end{array}$

Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$ b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.

Positive numbers $x,y$ satisfy equality $$x^3+y^3=x-y$$ Prove that $$x^2+y^2<1$$

For a positive integer $n$, we say an $n$-[i]shuffling[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$. Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3\}$, \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\] For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions. Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)-1}{q-1}\right)\right).\] Pick three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$ uniformly at random from the set of all $n$-shufflings. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\sigma_2,\sigma_3)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$.

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$: $p(5x)^2-3=p(5x^2+1)$ such that: $a) p(0)\neq 0$ $b) p(0)=0$

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties: (i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$. (ii) For all $S$, $T \in \mathcal{X}$, there hold \[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \] where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$. [i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

Find $ax^5 + by^5$ if the real numbers $a$, $b$, $x$, and $y$ satisfy the equations \begin{eqnarray*} ax + by &=& 3, \\ ax^2 + by^2 &=& 7, \\ ax^3 + by^3 &=& 16, \\ ax^4 + by^4 &=& 42. \end{eqnarray*}

Find all triplets $(x, y, z)$ of real numbers for which $$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$

Oleksii wrote some \( 2n \) (\( n > 1 \)) consecutive positive integers on the board. After that, he grouped these numbers into pairs in some way, and within each pair, he multiplied the two numbers together. He then wrote the resulting \( n \) products on the board instead of the original numbers. Afterward, Anton wrote down the difference between the largest and the smallest of the numbers Oleksii wrote. Oleksii wants Anton to write the smallest possible number. What is the smallest number that can be written? [i]Proposed by Oleksii Masalitin, Anton Trygub[/i]

Prove the following statement: If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real positve roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.

Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$