Define the sequence $(w_n)_{n\ge0}$ by the recurrence relation $$w_{n+2}=2w_{n+1}+3w_n,\enspace\enspace w_0=1,w_1=i,\enspace n=0,1,\ldots$$ (1) Find the general formula for $w_n$ and compute the first $9$ terms. (2) Show that $|\Re w_n-\Im w_n|=1$ for all $n\ge1$. [i]Proposed by Ovidiu Bagdasar[/i]

Determine the smallest value of $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers, so that $x^3 + y^3 + z^3 -3xyz = 1.$

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

Given two fractions $a/b$ and $c/d$ we define their [i]pirate sum[/i] as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$. Given an integer $n \ge 3$, initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$. At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.

Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.

Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression: $E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

The sequence $(a_n)$ of real numbers is defined as follows: \[a_1=1, \qquad a_2=2, \quad \text{and} \quad a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.\] Prove that for $n \geq 3$, $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$, where $[x]$ denotes the integer $p$ such that $p \leq x < p + 1$.

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and \[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$. Prove that they are converges and find their limits.

Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$; $f(x,y) = f(y,x)$; $f(x,1)=x$; $f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$

Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy: (i) $f$ is strictly increasing, (ii) $f(x) > -1/x$ for all $x > 0$, (iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$. Determine $f(1)$.

Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1\le i\le n$. Prove that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,\] and show that the statement is sharp, i.e. for an arbitrary $c>\frac29$ it is possible that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.\] [i]Submitted by Péter Pál Pach, Budapest, based on an OKTV problem[/i]

[b]p1.[/b] How many ordered triples of integers $(a, b, c)$ where $1 \le a, b, c \le 10$ are such that for every natural number, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? [b]p2.[/b] Find the smallest integer $n$ such that we can cut a $n \times n$ grid into $5$ rectangles with distinct side lengths in $\{1, 2, 3..., 10\}$. Every value is used exactly once. [b]p3.[/b] A plane is flying at constant altitude along a circle of radius $12$ miles with center at a point $A$.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point $A$, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$ $$f(m + f(n)) = f(m) - n.$$

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$. [i]Proposed by Tobi Moektijono, Indonesia[/i]

Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.