You are given an odd number $n\ge 3$. For every pair of integers $(i, j)$ with $1\le i \le j \le n$ there is a domino, with $i$ written on one its end and with $j$ written on another (there are $\frac{n(n+1)}{2}$ domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put $k$ dominos in this way, got bored and went away. After this Anton came to see this $k$ dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of $k$ is this possible? [i]Oleksii Masalitin[/i]

In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.

Let $\vartriangle ABC$ be acute with orthocenter $H$. Let $X$ be a point on $BC$ such that $B-X-C$. Let $\Gamma$ be the circumscribed circle of $\vartriangle BHX$ and $\Gamma_2$ be the circumscribed circle of $\vartriangle CHX$. Let $E$ be the intersection of $AB$ with $\Gamma$ , and $D$ be the intersection of $AC$ with $\Gamma_2$. Let $L$ be the intersection of line $HD$ with $\Gamma$ and $J$ be the intersection of line $EH$ with $\Gamma_2$. Prove that points $L$, $X$, and $J$ are collinear. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that \[ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. \]

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$

Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.

Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

As shown in figure , a circle of radius $1$ passes through vertex $A$ of $\vartriangle ABC$ and is tangent to the side $BC$ at the point $D$ , intersect sides $AB$ and $AC$ at points $E$ and $F$ respectively . Also$ EF$ bisects $\angle AFD$, and $\angle ADC = 80^o$ , Is there a triangle that satisfies the condition, so that $\frac{AB+BC+CA}{AD^2}$ is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition. [img]https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png[/img]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.

Let's construct a family $\{K_n\}$ of geometric figures following the pattern shown in pictures: [center][img]https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png[/img][/center] where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures $K_n$, for integer $n \ge 0$ . Then, we denote by $P_n$ and $A_n$ the perimeter and area of the figure $K_n$, respectively. If the side of square to build $K_0$ measures $x$: (a) Calculate $P_0$ and $A_0$ (in terms of the length $x$). (b) Find an explicit formula for $P_n$, and for $A_n$, in terms of $x$ and of $n$. Simplify your answers. (c) If $P_{2022} = A_{2022}$, find the measure of the six sides of the figure $K_0$, in its simplest form.

For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 23 \qquad\textbf{(E)}\ 30 $

Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

$6$ people stand in a circle with water guns. Each person randomly selects another person to shoot. What is the probability that no pair of people shoots at each other?

Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$. Determine the value of $\frac{BZ}{ZC}$.

If the line $L$ in the $xy$-plane has half the slope and twice the y-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: $ \textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad \textbf{(B)}\ y = \frac{4}{3} x + 2 \qquad \textbf{(C)}\ y = \frac{1}{3} x + 4 \qquad\\ \textbf{(D)}\ y = \frac{4}{3} x + 4 \qquad \textbf{(E)}\ y = \frac{1}{3} x + 2 $

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.

Let $p\ge 5$ be a prime number. Consider the function given by the formula $$f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.$$ Let $A_k$ denote the set of all these permutations $(a_1,..., a_p)$ of the set $\{1,..., p\}$, for integer number $f (a_1,..., a_p) - k$ is divisible by $p$ and $a_i \ne i$ for all $i \in \{1,..., p\}$. Prove that the sets $A_1$ and $A_4$ have the same number of elements.

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.