Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.

Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? $\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $

Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.

Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\]

How many cubics in the form $x^3 -ax^2 + (a+d)x -(a+2d)$ for integers $a,d$ have roots that are all non-negative integers?

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$

For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.

$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that $$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$

[u]Algebra Round[/u] [b]p1.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p2.[/b] $f(x) = x^2 + bx + c$ is a function with the property that the $x$-coordinate of the vertex is equal to the positive difference of the two roots of $f(x)$. Given that $c = 48$, compute $b$. [b]p3.[/b] Suppose we have a function $f(x)$ such that $f(x)^2 = f(x - 5)f(x + 5)$ for all integers $x$. Given that $f(1) = 1$ and $f(16) = 8$, what is $f(2016)$? [b]p4.[/b] Suppose that we have the following set of equations $$\log_2 x + \log_3 x + \log_4 x = 20$$ $$\log_4 y + \log_9 y + \log_{16} y = 16$$ Compute $\log_x y$. [b]p5.[/b] Let $\{a_n\}$ be the arithmetic sequence defined as $a_n = 2(n - 1) + 6$ for all $n \ge 1$. Compute $$\sum^{\infty}_{i=1} \frac{1}{a_ia_{i+2}}.$$ [b]p6.[/b] Let $a, b, c, d, e, f$ be non-negative real numbers. Suppose that $a + b + c + d + e + f = 1$ and $ad + be + cf \ge \frac{1}{18} $. Find the maximum value of $ab + bc + cd + de + ef + fa$. [b]p7.[/b] Let f be a continuous real-valued function defined on the positive real numbers. Determine all $f$ such that for all positive real $x, y$ we have $f(xy) = xf(y) + yf(x)$ and $f(2016) = 1$. [b]p8.[/b] Find the maximum of the following expression: $$21 cos \theta + 18 sin \theta sin \phi + 14 sin \theta cos \phi $$ [b]p9.[/b] $a, b, c, d$ satisfy the following system of equations $$ab + c + d = 13$$ $$bc + d + a = 27$$ $$cd + a + b = 30$$ $$da + b + c = 17.$$ Compute the value of $a + b + c + d$. [b]p10.[/b] Define a sequence of numbers $a_{n+1} = \frac{(2+\sqrt3)a_n+1}{(2+\sqrt3)-a_{n}}$ for $n > 0$, and suppose that $a_1 = 2$. What is $a_{2016}$? [u]Algebra Tiebreakers[/u] [b]Tie 1.[/b] Mark takes a two digit number $x$ and forms another two digit number by reversing the digits of $x$. He then sums the two values, obtaining a value which is divisible by $13$. Compute the smallest possible value of $x$. [b]Tie 2.[/b] Let $p(x) = x^4 - 10x^3 + cx^2 - 10x + 1$, where $c$ is a real number. Given that $p(x)$ has at least one real root, what is the maximum value of $c$? [b]Tie 3.[/b] $x$ satisfies the equation $(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0$. Compute the largest possible value of $|x|$. PS. You should use hide for answers.

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

For $n = 1,2,3,...$. $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

Prove the equality:\\ $\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}$ .

Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations : $$x^2+xy+y^2=57$$ $$y^2+yz+z^2=84$$ $$z^2+zx+x^2=111$$ What is the value of $xy+3yz+5zx$? [i](maphybich)[/i]

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

Juan found an easy (but wrong) way to simplify fractions. He proposes to simplify a fraction $\frac{M}{N}$ , where $M <N$ are two natural numbers, erase simultaneously the equal digits in the numerator and denominator. For instance, $\frac{12356}{5789}$ transforms after simplification of Juan in $\frac{126}{789}$. Find out if there is at least one fraction $\frac{M}{N}$, with $10 <M <N <100$ for which this method gives a correct result.

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.