The line passing through $B$ is perpendicular to the side $AC$ at $E$. This line meets the circumcircle of $\triangle ABC$ at $D$. The foot of the perpendicular from $D$ to the side $BC$ is $F$. If $O$ is the center of the circumcircle of $\triangle ABC$, prove that $BO$ is perpendicular to $EF$.

$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and $N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively . It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$. (Grigory Filippovsky)

The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$

Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$.

Let $ABC$ be a triangle with $AB=c$ , $BC=a$ and $AC=b$. If $ x,y\in\mathbb{R}$ satisfy $ \frac{1}{x} +\frac{1}{y+z} = \frac{1}{a} $ , $ \frac{1}{y} +\frac{1}{x+z} = \frac{1}{b} $ , $ \frac{1}{z} +\frac{1}{y+x} = \frac{1}{c} $ . Prove that the following equality holds $ x(p-a) + y(p-b) + z(p-c) = 3r^2 + 12R*r , $ Where $p$ is semi-perimeter, $R$ is the circumradius and $r$ is the inradius.

Ana and Beta play a turn-based game on a $m \times n$ board. Ana begins. At the beginning, there is a stone in the lower left square and the objective is to move it to the upper right corner. A move consists of the player moving the stone to the right or up as many squares as the player wants. Find all the values ​​of $(m, n)$ for which Ana can guarantee victory.

In $\triangle ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that each of the quadrilaterals $AFDE$, $BDEF$, and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$.

Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write \[n =\sum_{m\in O_n}2^m;\] the valid numbers are the elements $m$ of $O_n$. That way, for example, after Arnaldo says $42= 2^5 + 2^3 + 2^1$, Bernaldo must respond with $5$, $3$ or $1$. We define the sets $A,B\subset \mathbb{N}$ in the following way. We have $n\in A$ iff Arnaldo, saying $n$ in his first turn, has a winning strategy; analogously, we have $n\in B$ iff Bernaldo has a winning strategy if Arnaldo says $n$ during his first turn. This way, \[A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}\] Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=|A\cap \{0,1,\cdots,n-1\}|$. For example, $f(8) = 2$ and $f(11)=4$. Find \[\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}\]

A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is $58$. Find $a$ and $b$.

A rocket is launched and reaches $120$ m in height; in the fall he loses $60$ m, then it recovers $40$ m, loses $ 30 $ again, gains $24$, loses $20$, etc. If the process continues indefinitely, at what height does it tend to stabilize?

Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

How many of the following capital English letters look the same when rotated $180^\circ$ about their center? [center]A B C D E F G H I J K L M N O P Q R S T U V W X Y Z[/center] [i]Proposed by William Yue[/i]

Given a tetrahedron $PABC$, draw the height $PH$ from vertex $P$ to $ABC$. From point $H$, draw perpendiculars $HA’,HB’,HC’$ to the lines $PA,PB,PC$. Suppose the planes $ABC$ and $A’B’C’$ intersects at line $\ell$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $OH\perp \ell$.

The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is: $\textbf{(A)}\ 30\qquad \textbf{(B)}\ \frac{80}{3}\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ \frac{45}{2}\qquad \textbf{(E)}\ 20$

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

The integers $a, b, c$ are not $0$ such that $\frac{a}{b + c^2}=\frac{a + c^2}{b}$. Prove that $a + b + c \le 0$.

Benji has a $2\times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?

Prove that for every integer $ n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n\minus{}1)<n^n.$

Let $O$ be the midpoint of the axis of a right circular cylinder. Let $A$ and $B$ be diametrically opposite points of one base, and $C$ a point of the other base circle that does not belong to the plane $OAB$. Prove that the sum of dihedral angles of the trihedral $OABC$ is equal to $2\pi$.

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and $DM=CN$ then prove that $AC=BD$. S. Berlov

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

[b]p5.[/b] Find the number of three-digit integers that contain at least one $0$ or $5$. The leading digit of the three-digit integer cannot be zero. [b]p6.[/b] What is the sum of the solutions to $\frac{x+8}{5x+7} =\frac{x+8}{7x+5}$ [b]p7.[/b] Let $BC$ be a diameter of a circle with center $O$ and radius $4$. Point $A$ is on the circle such that $\angle AOB = 45^o$. Point $D$ is on the circle such that line segment$ OD$ intersects line segment $AC$ at $E$ and $OD$ bisects $\angle AOC$. Compute the area of $ADE$, which is enclosed by line segments $AE$ and $ED$ and minor arc $AD$. [b]p8. [/b] William is a bacteria farmer. He would like to give his fiance$ 2021$ bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those $2021$ bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals? PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. ) ( F . L . Nazarov)

Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$. [i]Linus Hamilton.[/i]