Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$ satisfy $|z_3|=|z_4|=1$

For all sets $A$ of complex numbers, let $P(A)$ be the product of the elements of $A$. Let $S_z = \{1, 2, 9, 99, 999, \frac{1}{z},\frac{1}{z^2}\}$, let $T_z$ be the set of nonempty subsets of $S_z$ (including $S_z$), and let $f(z) = 1 + \sum_{s\in T_z} P(s)$. Suppose $f(z) = 6125000$ for some complex number $z$. Compute the product of all possible values of $z$.

Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.

In a graph with $n$ vertices every two vertices are connected by a unique path. For each two vertices $u$ and $v$, let $d(u, v)$ denote the distance between $u$ and $v$, i.e. the number of edges in the path connecting these two vertices, and $\deg(u)$ denote the degree of a vertex $u$. Let $W$ be the sum of pairwise distances between the vertices, and $D$ the sum of weighted pairwise distances: $\sum_{\{u, v\}}(\deg(u)+\deg(v))d(u, v)$. Prove that $D=4W-n(n-1)$.

You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?

The three masses shown in the accompanying diagram are equal. The pulleys are small, the string is lightweight, and friction is negligible. Assuming the system is in equilibrium, what is the ratio $a/b$? The figure is not drawn to scale! [asy] size(250); dotfactor=10; dot((0,0)); dot((15,0)); draw((-3,0)--(25,0),dashed); draw((0,0)--(0,3),dashed); draw((15,0)--(15,3),dashed); draw((0,0)--(0,-15)); draw((15,0)--(15,-10)); filldraw(circle((0,-16),1),lightgray); filldraw(circle((15,-11),1),lightgray); draw((0,0)--(4,-4)); filldraw(circle((4.707,-4.707),1),lightgray); draw((15,0)--(5.62,-4.29)); draw((0.5,3)--(14.5,3),Arrows(size=5)); label(scale(1.2)*"$a$",(7.5,3),1.5*N); draw((2.707,-4.707)--(25,-4.707),dashed); draw((25,-0.5)--(25,-4.2),Arrows(size=5)); label(scale(1.2)*"$b$",(25,-2.35),1.5*E); [/asy] (A) $1/2$ (B) $1$ (C) $\sqrt{3}$ (D) $2$ (E) $2\sqrt{3}$

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

Petya and Kolya play the following game: they take turns changing one of the coefficients $a$ or $b$ of the quadratic trinomial $f = x^2 + ax + b$: Petya is on $1$, Kolya is on $1$ or $3$. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds $a$ and $b$ regardless of Petya's game? [hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?[/hide]

Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.

Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$ Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$

A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.

Let $ABCD$ be a circumscribed quadrilateral with incircle $\gamma$. Let $AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L$. Let a circle with diameter $KL$ intersect $\gamma$ at one of the points $X$. Prove that $(EXF)$ is tangent to $\gamma$.

Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010?

Let $\triangle ABC$ be an equilateral triangle. If $0 < r < 1$, let $D_r$ be the point on $\overline{AB}$ such that $AD_r = r \cdot AB$, let $E_r$ be the point on $\overline{BC}$ such that $BE_r = r \cdot BC$, and let $P_r$ be the point where $\overline{AE_r}$ and $\overline{CD_r}$ intersect. Prove that the set of points $P_r$ (over all $0 < r < 1$) lie on a circle.

Find the number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan[/i]

What fraction of the Earth's volume lies above the $45$ degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.