The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series \[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]

On a faded piece of paper it is possible to read the following: \[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\] Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$? We assume that all polynomials in the statement have only integer coefficients.

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that: $$f(xy+f(x))=xf(y)$$ for all $x,y\in\mathbb{R}$.

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?

Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$ [i]Proposed by Kyle Lee[/i]

Let $n$ be a positive integer and let $f_1(x), f_2(x) \dots f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$. Determine the largest and smallest possible values of $|S|$ in terms of $n$. (A function $f(x)$ is affine if it is of form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$)

Find all positive real numbers $(x,y,z)$ such that: $$x = \frac{1}{y^2+y-1}$$ $$y = \frac{1}{z^2+z-1}$$ $$z = \frac{1}{x^2+x-1}$$

For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that \[b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.\]

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$.

Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$

Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.

Find all $(x,y,z)\in\mathbb R^3$ such that $$x+y+z=xy+yz+zx=3$$ [i]Proposed by Ezra Guerrero, Francisco Morazan[/i]

Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$. Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.

Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$. (b) Is this sequence periodic? And why?

Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$ [i]Dimitar Trenevski[/i]

Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ . $(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$ $(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$ for the integer $k\ge 3.$