$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

[b]p1.[/b] One out of $12$ coins is counterfeited. It is known that its weight differs from the weight of a valid coin but it is unknown whether it is lighter or heavier. How to detect the counterfeited coin with the help of four trials using only a two-pan balance without weights? [b]p2.[/b] Below a $3$-digit number $c d e$ is multiplied by a $2$-digit number $a b$ . Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits. $\begin{tabular}{ccccc} & & c & d & e \\ x & & & a & b \\ \hline & & f & e & g \\ + & c & d & e & \\ \hline & b & b & c & g \\ \end{tabular}$ [b]p3.[/b] Find all integer $n$ such that $\frac{n + 1}{2n - 1}$is an integer. [b]p4[/b]. There are several straight lines on the plane which split the plane in several pieces. Is it possible to paint the plane in brown and green such that each piece is painted one color and no pieces having a common side are painted the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $n$, let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$. Find the value of $$A(T(2021))$$

The sequence $ a_n$ is defined as follows: $ a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}$. Prove that all the terms of the sequence are integers.

Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ $f(f(x)f(y)+y)=f(x)y+f(y-x+1)$ For all $x,y \in \mathbb{R}$

Let $a$ and $b$ be positive integers with $a>b$. Suppose that $$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$ is an integer. (a) Must $\sqrt{a}$ be an integer? (b) Must $\sqrt{b}$ be an integer?

Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations: \begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for $ S \equal{} abc \plus{} \frac {1}{abc}$

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

[b]p1.[/b] Given any four digit number $X = \underline{ABCD}$, consider the quantity $Y(X) = 2 \cdot \underline{AB}+\underline{CD}$. For example, if $X = 1234$, then $Y(X) = 2 \cdot 12+34 = 58$. Find the sum of all natural numbers $n \le 10000$ such that over all four digit numbers $X$, the number $n$ divides $X$ if and only if it also divides $Y(X)$. [b]p2.[/b] A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes $5$ minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes $10$ minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes $15$ seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)? [b]p3.[/b] One of the bases of a right triangular prism is a triangle $XYZ$ with side lengths $XY = 13$, $YZ = 14$, $ZX = 15$. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing $\overline{YZ}$ cuts the prism and the sphere, giving rise to a cross-section of area $A$ for the prism and area $15\pi$ for the sphere. Find the sum of all possible values of $A$. [b]p4.[/b] Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length $x$, Brian a branch of length $y$, and Christine a branch of length $z$, and the lengths follow the condition that $x+y+z = 2$. Let $m$ and $n$ be the minimum and maximum possible values of $xy+yz+xz-xyz$, respectively. What is $m+n$? [b]p5.[/b] Let $S := MATHEMATICSMATHEMATICSMATHE...$ be the sequence where $7$ copies of the word $MATHEMATICS$ are concatenated together. How many ways are there to delete all but five letters of $S$ such that the resulting subsequence is $CHMMC$? [b]p6.[/b] Consider two sequences of integers $a_n$ and $b_n$ such that $a_1 = a_2 = 1$, $b_1 = b_2 = 1$ and that the following recursive relations are satisfied for integers $n > 2$: $$a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},$$ $$b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.$$ Determine the value of $$\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.$$ [b]p7.[/b] Suppose $ABC$ is a triangle with circumcenter $O$. Let $A'$ be the reflection of $A$ across $\overline{BC}$. If $BC =12$, $\angle BAC = 60^o$, and the perimeter of $ABC$ is $30$, then find $A'O$. [b]p8.[/b] A class of $10$ students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a $\frac12$ probability of putting a slip of paper with their own name into the box and a $\frac12$ probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected. [b]p9.[/b] Let $a$ and $b$ be positive integers, $a > b$, such that $6! \cdot 11$ divides $x^a -x^b$ for all positive integers $x$. What is the minimum possible value of $a+b$? [b]p10.[/b] Find the number of pairs of positive integers $(m,n)$ such that $n < m \le 100$ and the polynomial $x^m+x^n+1$ has a root on the unit circle. [b]p11.[/b] Let $ABC$ be a triangle and let $\omega$ be the circle passing through $A$, $B$, $C$ with center $O$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are drawn tangent to $\omega$ at $A$, $B$, $C$ respectively. The intersections of these lines form a triangle $XYZ$ where $X$ is the intersection of $\ell_B$ and $\ell_C$, $Y$ is the intersection of $\ell_C$ and $\ell_A$, and $Z$ is the intersection of $\ell_A$ and $\ell_B$. Let $P$ be the intersection of lines $\overline{OX}$ and $\overline{YZ}$. Given $\angle ACB = \frac32 \angle ABC$ and $\frac{AC}{AB} = \frac{15}{16}$ , find $\frac{ZP}{YP}$. [b]p12.[/b] Compute the remainder when $$\sum_{1\le a,k\le 2021} a^k$$ is divided by $2022$ (in the above summation $a,k$ are integers). [b]p13.[/b] Consider a $7\times 2$ grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )? [b]p14.[/b] Let $ABC$ be a triangle with $AB = 8$, $BC = 10$, and $CA = 12$. Denote by $\Omega_A$ the $A$-excircle of $ABC$, and suppose that $\Omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $F$ and $E$, respectively. Line $\ell \ne \overline{BC}$ is tangent to $\Omega_A$ and passes through the midpoint of $\overline{BC}$. Let $T$ be the intersection of $\overline{EF}$ and $\ell$. Compute the area of triangle $ATB$. [b]p15.[/b] For any positive integer $n$, let $D_n$ be the set of ordered pairs of positive integers $(m,d)$ such that $d$ divides $n$ and gcd$(m,n) = 1$, $1 \le m \le n$. For any positive integers $a$, $b$, let $r(a,b)$ be the non-negative remainder when $a$ is divided by $b$. Denote by $S_n$ the sum $$S_n = \sum_{(m,d)\in D_n} r(m,d).$$ Determine the value of $S_{396}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b$ be given two real number with $a \ne 0$. Find all polynomials $P$ with real coefficients such that $x P(x - a) = (x - b)P(x)$ for all $x\in R$

We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.

Find all positive integers $ a$ and $ b$ for which \[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]

Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.

Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.

All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.