On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $

Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i\equal{}1}^{300} a_i \equal{} 1$. Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.

Let $HOPKINS$ be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than $180^{\circ}$) with area $1876$ such that all of its side lengths are greater than $5$, $OP=20$, and $KI=22$. Arcs with radius $2$ are drawn inside $HOPKINS$ with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside $HOPKINS$ but outside the sectors.

Let $G$ be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs $(n,k)$, $n\ge 3$, such that: a) For all graph $G$ with $n$ vertex and $k$ edges, it is always possible to perform a series of switching process so that all edges are eventually blue. b) There exist a graph $G$ with $n$ vertex and $k$ edges and it is possible to perform a series of switching process so that all edges are eventually blue.

Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.

Suppose $n\geq 3$ is an integer. There are $n$ grids on a circle. We put a stone in each grid. Find all positive integer $n$, such that we can perform the following operation $n-2$ times, and then there exists a grid with $n-1$ stones in it: $\bullet$ Pick a grid $A$ with at least one stone in it. And pick a positive integer $k\leq n-1$. Take all stones in the $k$-th grid after $A$ in anticlockwise direction. And put then in the $k$-th grid after $A$ in clockwise direction.

Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$ $(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0$ $(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0$

Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

A [b][u]word[/u][/b] is formed by a number of letters of the alphabet. We show words with capital letters. A [b][u]sentence[/u][/b] is formed by a number of words. For example if $A=aa$ and $B=ab$ then the sentence $AB$ is equivalent to $aaab$. In this language, $A^n$ indicates $\underbrace{AA \cdots A}_{n}$. We have an equation when two sentences are equal. For example $XYX=YZ^2$ and it means that if we write the alphabetic letters forming the words of each sentence, we get two equivalent sequences of alphabetic letters. An equation is [b][u]simplified[/u][/b], if the words of the left and the right side of the sentences of the both sides of the equation are different. Note that every word contains one alphabetic letter at least. $\text{a})$We have a simplified equation in terms of $X$ and $Y$. Prove that both $X$ and $Y$ can be written in form of a power of a word like $Z$.($Z$ can contain only one alphabetic letter). $\text{b})$ Words $W_1,W_2,\cdots , W_n$ are the answers of a simplified equation. Prove that we can produce these $n$ words with fewer words. $\text{c})$ $n$ words $W_1,W_2,\cdots , W_n$ are the answers of a simplified system of equations. Define graph $G$ with vertices ${1,2 \cdots ,n}$ such that $i$ and $j$ are connected if in one of the equations, $W_i$ and $W_j$ be the two words appearing in the right side of each side of the equation.($\cdots W_i = \cdots W_j$). If we denote by $c$ the number of connected components of $G$, prove that these $n$ words can be produced with at most $c$ words. [i]Proposed by Mostafa Einollah Zadeh Samadi[/i]

In a round-robin tournament with $n$ players $P_1$, $P_2$, $\ldots$, $P_n$ (where $n > 1$), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $w_r$ and $l_r$ be the number of games won and lost, respectively, by $P_r$. Show that \[ \sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2. \]

Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$. (a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle. (b) Prove that the line $ST$ is parallel to the side $BC$. (Karl Czakler)

Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.

There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$, we can move one to ${B_2}$. If there is at least $1$ ball in ${B_n}$, we can move one to ${B_{n - 1}}$. If there are at least $2$ balls in ${B_k}$, $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$, and one to ${B_{k + 1}}$. Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.

A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$ [i]Proposed by Alex Li[/i]

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

Determine the number of 10-letter strings consisting of $A$s, $B$s, and $C$s such that there is no $B$ between any two $A$s.

The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1, O_2$, and $O_3$ of the triangles $LBC, LCD$, and $LDA$ were marked. Then the whole configuration except for points $H, O_1, O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\ y^2 - 3z + p = x, \\ z^2 - 3x + p = y \end{cases}$ with real parameter $p$. a) For $p \ge 4$, solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$. (Jaroslav Svrcek)

To the nearest degree, find the measure of the largest angle in a triangle with side lengths $3$, $5$, and $7$.

Along a track for cross-country skiing, $1000$ seats are placed in a row and numbered in order from $1$ to $1000$. By mistake, $n$ tickets were sold, $100 < n < 1000$, each with one of the numbers $1,2,..., 100$ printed on it. Also for each number $1,2,..., 100$ there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These $n$ spectators arrive one at a time. Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets. (A Shen)