In a convex quadrilateral $ABCD$ we have $AB\cdot CD=BC\cdot DA$ and $2\angle A+\angle C=180^\circ$. Point $P$ lies on the circumcircle of triangle $ABD$ and is the midpoint of the arc $BD$ not containing $A$. It is known that the point $P$ lies inside the quadrilateral $ABCD$. Prove that $\angle BCA=\angle DCP$ [i]Proposed by S. Berlov[/i]

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

Find all real $a,b,c,d,e,f$ that satisfy the system $4a = (b + c + d + e)^4$ $4b = (c + d + e + f)^4$ $4c = (d + e + f + a)^4$ $4d = (e + f + a + b)^4$ $4e = (f + a + b + c)^4$ $4f = (a + b + c + d)^4$

It is given a convex quadrilateral $ ABCD$ in which $ \angle B\plus{}\angle C < 180^0$. Lines $ AB$ and $ CD$ intersect in point E. Prove that $ CD*CE\equal{}AC^2\plus{}AB*AE \leftrightarrow \angle B\equal{} \angle D$

A sequence of $2n + 1$ non-negative integers $a_1, a_2, ..., a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, ..., 2n + 1$. Starting from this initial position, we repeat the following sequence of steps, as long as it's possible: [i]Step 1[/i]: Add up the numbers written on all the cells, denote the sum as $s$. [i]Step 2[/i]: If $s$ is equal to $0$ or if it is larger than the current number of cells, the process terminates. Otherwise, remove the $s$-th cell, and shift shift all cells that are to the right of it one position to the left. Then go to Step 1. Example: $(1, 0, 1, \underline{2}, 0) \rightarrow (1, \underline{0}, 1, 0) \rightarrow (1, \underline{1}, 0) \rightarrow (\underline{1}, 0) \rightarrow (0)$. A sequence $a_1, a_2,. . . , a_{2n+1}$ of non-negative integers is called balanced, if at the end of this process there’s exactly one cell left, and it’s the cell that was initially enumerated by $(n + 1)$, i.e. the cell that was initially in the middle. Find the total number of balanced sequences as a function of $n$. [i]Proposed by Viktor Simjanoski, North Macedonia[/i]

Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.

There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$

Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$.

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+yf(x))=f(x)+xf(y) \quad \text{for all}\ x,y \in\mathbb{R}\]

Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$

Consider an acute-angled triangle $ABC$ with $AB < AC$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AB$, respectively. The circle with diameter $AB$ intersects the lines $BC, AM$ and $AC$ at $D, E$, and $F$, respectively. Let $G$ be the midpoint of $FC$. Prove that the lines $NF, DE$ and $GM$ are concurrent. [i]Michal Pecho[/i]

[u]Round 5[/u] [b]p13.[/b] Determine the number of different circular bracelets can be made with $7$ beads, all either colored red or black. [b]p14.[/b] The product of $260$ and $n$ is a perfect square. The $2020$th least possible positive integer value of $n$ can be written as$ p^{e_1}_1 \cdot p^{e_2}_2\cdot p^{e_3}_3\cdot p^{e_4}_4$ . Find the sum $p_1 +p_2 +p_3 +p_4 +e_1 +e_2 +_e3 +e_4$. [b]p15.[/b] Let $B$ and $C$ be points along the circumference of circle $\omega$. Let $A$ be the intersection of the tangents at $B$ and $C$ and let $D \ne A$ be on $\overrightarrow{AC}$ such that $AC =CD = 6$. Given $\angle BAC = 60^o$, find the distance from point $D$ to the center of $\omega$. [u]Round 6[/u] [b]p16.[/b] Evaluate $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$. [b]p17.[/b] Let $n(A)$ be the number of elements of set $A$ and $||A||$ be the number of subsets of set $A$. Given that $||A||+2||B|| = 2^{2020}$, find the value of $n(B)$. [b]p18.[/b] $a$ and $b$ are positive integers and $8^a9^b$ has $578$ factors. Find $ab$. [u]Round 7[/u] [b]p19.[/b] Determine the probability that a randomly chosen positive integer is relatively prime to $2019$. [b]p20.[/b] A $3$-by-$3$ grid of squares is to be numbered with the digits $1$ through $9$ such that each number is used once and no two even-numbered squares are adjacent. Determine the number of ways to number the grid. [b]p21.[/b] In $\vartriangle ABC$, point $D$ is on $AC$ so that $\frac{AD}{DC}= \frac{1}{13}$ . Let point $E$ be on $BC$, and let $F$ be the intersection of $AE$ and $BD$. If $\frac{DF}{FB}=\frac{2}{7}$ and the area of $\vartriangle DBC$ is $26$, compute the area of $\vartriangle F AB$. [u]Round 8[/u] [b]p22.[/b] A quarter circle with radius $1$ is located on a line with its horizontal base on the line and to the left of the vertical side. It is then rolled to the right until it reaches its original orientation. Determine the distance traveled by the center of the quarter circle. [b]p23.[/b] In $1734$, mathematician Leonhard Euler proved that $\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+...$. With this in mind, calculate the value of $\frac11-\frac14+\frac19-\frac{1}{16}+...$ (the series obtained by negating every other term of the original series). [b]p24.[/b] Billy the biker is competing in a bike show where he can do a variety of tricks. He knows that one trick is worth $2$ points, $1$ trick is worth $3$ points, and 1 is worth $5$ points, but he doesn’t remember which trick is worth what amount. When it’s Billy’s turn to perform, he does $6$ tricks, randomly choosing which trick to do. Compute the sum of all the possible values of points that Billy could receive in total. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the line $BC$ at $A_1$. $C_2$ is the reflection of $C_1$ in the midpoint of $AB$, and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$. The lines $B_2C_2$ and $BC$ intersect at $A_2$. Prove that $$\frac{sen \, \, B_1A_1C}{sen\, \, C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$ [img]https://cdn.artofproblemsolving.com/attachments/3/8/774da81495df0a0f7f2f660ae9f516cf70df06.png[/img]

An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.

Alice and Bob take turns removing balls from a bag containing $10$ black balls and $10$ white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls which Bob has can be expressed as $a/b$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

Let $\vartriangle ABC$ be a triangle with orthocenter $H$, and let $M$, $N$ be the midpoints of $BC$ and $AH$ respectively. Suppose $Q$ is a point on $(ABC)$ such that $\angle AQH = 90^o$. Show that $MN$, the circumcircle of $QNH$, and the $A$-symmedian concur. Note: the $A$-symmedian is the reflection of line $AM$ in the bisector of angle $\angle BAC$.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

Find all triples $(a,b,c)$ of nonzero integers for which \[(1-a)(1-b)(1-c)=(1+a)(1+b)(1+c).\]

On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions: (A) Gives a candy to the student sitting on his left or to the student sitting on his right. (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right. At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions. (Two distributions are different if there is a student who has a different number of candy in each of these distributions.) （Forgive my poor English）

$10$ people are sitting at a round table. There are some nuts in front of each of them, $100$ nuts altogether. After a certain signal each person passes some of his nuts to the person sitting to his right . If he has an even number of nuts, he passes half of them; otherwise he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone will have exactly $10$ nuts. (A Shapovalov)

Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values ​​of $n$?

Let $a$ be the root of the equation $x^3-x-1=0$. Find an equation of the third degree with integer coefficients whose root is $a^3$.