Let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, of an equilateral triangle $ABC$. If $AC' = 2C'B$, $BA' = 2A'C$ and $CB' = 2B'A$, prove that the lines $AA'$, $BB'$ and $CC'$ enclose a triangle whose area is $1/7$ that of $ABC$.

Four teams $A$, $B$, $C$ and $D$ play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are $12$ matches). In each matchif it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $3$ points and the loser gets $0$ points. At the end of the tournament the teams $A$, $B$ and $C$ have $8$ points each. Determine all possible points of team $D$.

The projection of the vertex $C$ of the rectangle $ABCD$ on the diagonal $BD$ is $E$. The projections of $E$ on $AB$ and $AD$ are $F$ and $G$ respectively. Prove that $$AF^{2/3} + AG^{2/3} = AC^{2/3}$$ .

Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$. 1. Find the distance between the centers of the circles(using $a$ and $b$). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

Let $ABC$ be a triangle with $m(\widehat A) = 70^\circ$ and the incenter $I$. If $|BC|=|AC|+|AI|$, then what is $m(\widehat B)$? $ \textbf{(A)}\ 35^\circ \qquad\textbf{(B)}\ 36^\circ \qquad\textbf{(C)}\ 42^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ \text{None of above} $

There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.

A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron. [b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before. [b]b)[/b] prove that the number four in previous part can't be replaced with three. [i]proposed by Kasra Alishahi[/i]

Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.

Let $n\geqslant 3$ be a positive integer and $N=\{1,2,\ldots,n\}$ and let $k>0$ be a real number. Let's associate each non-empty of $N{}$ with a point in the plane, such that any two distinct subsets correspond to different points. If the absolute value of the difference between the arithmetic means of the elements of two distinct non-empty subsets of $N{}$ is at most $k{}$ we connect the points associated with these subsets with a segment. Determine the minimum value of $k{}$ such that the points associated with any two distinct non-empty subsets of $N{}$ are connected by a segment or a broken line. [i]Cristi Săvescu[/i]

A blue square of side length $10$ is laid on top of a coordinate grid with corners at $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$. Red squares of side length $2$ are randomly placed on top of the grid, changing the color of a $2\times2$ square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by $\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.

In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$. [asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10); fill(Arc((0,0),1,10,90)--C--D--cycle,mediumgray); fill(Arc((0,0),1,190,270)--B--F--cycle,mediumgray); draw(unitcircle); draw(A--B--D^^A--E); label("$A$",A,S); label("$B$",B,W); label("$C$",C,SE); label("$\theta$",C,SW); label("$D$",D,NE); label("$E$",E,N); [/asy] A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is $ \textbf{(A)}\ \tan \theta = \theta\qquad\textbf{(B)}\ \tan \theta = 2\theta\qquad\textbf{(C)}\ \tan \theta = 4\theta\qquad\textbf{(D)}\ \tan 2\theta = \theta\qquad \\ \textbf{(E)}\ \tan \frac{\theta}{2} = \theta$

Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic. Proposed by:Y.Shcherbatov

[u]Round 5[/u] [b]p13.[/b] Given a (not necessarily simplified) fraction $\frac{m}{n}$ , where $m, n > 6$ are positive integers, when $6$ is subtracted from both the numerator and denominator, the resulting fraction is equal to $\frac45$ of the original fraction. How many possible ordered pairs $(m, n)$ are there? [b]p14.[/b] Jamesu's favorite anime show has $3$ seasons, with $12$ episodes each. For $8$ days, Jamesu does the following: on the $n^{th}$ day, he chooses $n$ consecutive episodes of exactly one season, and watches them in order. How many ways are there for Jamesu to finish all $3$ seasons by the end of these $8$ days? (For example, on the first day, he could watch episode $5$ of the first season; on the second day, he could watch episodes $11$ and $12$ of the third season, etc.) [b]p15.[/b] Let $O$ be the center of regular octagon $ABCDEFGH$ with side length $6$. Let the altitude from $O$ meet side $AB$ at $M$, and let $BH$ meet $OM$ at $K$. Find the value of $BH \cdot BK$. [u]Round 6[/u] [b]p16.[/b] Fhomas writes the ordered pair $(2, 4)$ on a chalkboard. Every minute, he erases the two numbers $(a, b)$, and replaces them with the pair $(a^2 + b^2, 2ab)$. What is the largest number on the board after $10$ minutes have passed? [b]p17.[/b] Triangle $BAC$ has a right angle at $A$. Point $M$ is the midpoint of $BC$, and $P$ is the midpoint of $BM$. Point $D$ is the point where the angle bisector of $\angle BAC$ meets $BC$. If $\angle BPA = 90^o$, what is $\frac{PD}{DM}$? [b]p18.[/b] A square is called legendary if there exist two different positive integers $a, b$ such that the square can be tiled by an equal number of non-overlapping $a$ by $a$ squares and $b$ by $b$ squares. What is the smallest positive integer $n$ such that an $n$ by $n$ square is legendary? [u]Round 7[/u] [b]p19.[/b] Let $S(n)$ be the sum of the digits of a positive integer $n$. Let $a_1 = 2019!$, and $a_n = S(a_{n-1})$. Given that $a_3$ is even, find the smallest integer $n \ge 2$ such that $a_n = an_1$. [b]p20.[/b] The local EMCC bakery sells one cookie for $p$ dollars ($p$ is not necessarily an integer), but has a special offer, where any non-zero purchase of cookies will come with one additional free cookie. With $\$27:50$, Max is able to buy a whole number of cookies (including the free cookie) with a single purchase and no change leftover. If the price of each cookie were $3$ dollars lower, however, he would be able to buy double the number of cookies as before in a single purchase (again counting the free cookie) with no change leftover. What is the value of $p$? [b]p21.[/b] Let circle $\omega$ be inscribed in rhombus $ABCD$, with $\angle ABC < 90^o$. Let the midpoint of side $AB$ be labeled $M$, and let $\omega$ be tangent to side $AB$ at $E$. Let the line tangent to $\omega$ passing through $M$ other than line $AB$ intersect segment $BC$ at $F$. If $AE = 3$ and $BE = 12$, what is the area of $\vartriangle MFB$? [u]Round 8[/u] [b]p22.[/b] Find the remainder when $1010 \cdot 1009! + 1011 \cdot 1008! + ... + 2018 \cdot 1!$ is divided by $2019$. [b]p23.[/b] Two circles $\omega_1$ and $\omega_2$ have radii $1$ and $2$, respectively and are externally tangent to one another. Circle $\omega_3$ is externally tangent to both $\omega_1$ and $\omega_2$. Let $M$ be the common external tangent of $\omega_1$ and $\omega_3$ that doesn't intersect $\omega_2$. Similarly, let $N$ be the common external tangent of $\omega_2$ and $\omega_3$ that doesn't intersect $\omega_1$. Given that $M$ and N are parallel, find the radius of $\omega_3$. [b]p24.[/b] Mana is standing in the plane at $(0, 0)$, and wants to go to the EMCCiffel Tower at $(6, 6)$. At any point in time, Mana can attempt to move $1$ unit to an adjacent lattice point, or to make a knight's move, moving diagonally to a lattice point $\sqrt5$ units away. However, Mana is deathly afraid of negative numbers, so she will make sure never to decrease her $x$ or $y$ values. How many distinct paths can Mana take to her destination? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949411p26408196]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $a, b, c$ and $d$ are positive real numbers such that $a + b + c + d = 2$, prove that $$\frac{a^2}{(a^2+1)^2}+\frac{b^2}{(b^2+1)^2}+\frac{c^2}{(c^2+1)^2}+\frac{d^2}{(d^2+1)^2} \le \frac{16}{25}$$

In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.

Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0, 0)$ and $(1, 0)$, and the other two vertices are two distinct lattice points $(a, b)$, $(c, d)$ with $0 \le a$, $c \le 59$ and $1 \le b$, $d \le 5.$ (A concave quadrilateral is a quadrilateral with an angle strictly larger than $180^o$. A lattice point is a point with both coordinates integers.)

Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //label("$A$", A, dir(point--A)); //label("$B$", B, dir(point--B)); //label("$C$", C, dir(point--C)); //label("$D$", D, dir(point--D)); label("$1/n$", (11/12,1), N, fontsize(9));[/asy]

Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. [i]Proposed by Nikola Velov, North Macedonia[/i]

The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let $p$ be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of $p$ and $s$.

Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon. [i]Daniel Jinga[/i]

Yan and Kirill play a game. At first Kirill says 4 numbers $x_1<x_2<x_3<x_4$, and then Yan says three pairwise different non zero numbers $a_1$, $a_2$ and $a_3$. For all $i$ from $1$ to $3$ they consider the quadratic trinomial $f_i(x)$ which has roots $x_i$ and $x_{i+1}$ and leading coefficient $a_i$, and construct on the plane the graphs of that trinomials. Yan wins if in every pair $(f_1(x),f_2(x))$ and $(f_2(x),f_3(x))$ their graphs intersect at exactly one point, and if in some pair graphs do not intersect or intersect at more than one point Kirill wins. Find which player can guarantee his win regardless of the actions of his opponent. [i]V. Kamianetski[/i]

If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].