Two particles move along the edges of equilateral triangle $ \triangle ABC$ in the direction \[ A\rightarrow B\rightarrow C\rightarrow A \]starting simultaneously and moving at the same speed. One starts at $ A$, and the other starts at the midpoint of $ \overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $ R$. What is the ratio of the area of $ R$ to the area of $ \triangle ABC$? $ \textbf{(A)}\ \frac {1}{16}\qquad \textbf{(B)}\ \frac {1}{12}\qquad \textbf{(C)}\ \frac {1}{9}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.

A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$. It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the segment $BC$. [i] (А. Кузнецов)[/i]

Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

In a set of objects, each has one of two colors and one of two shapes. There is at least one object of each color and at least one object of each shape. Prove that there exist two objects in the set that are different both in color and in shape.

For real numbers $a,b,c$ and positive number $\lambda$ such that three real roots $x_1,x_2,x_3$ of $f(x)=x^3+ax^2+bx+c$ satisfying: $(1) x_2-x_1=\lambda$; $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

The integers from $1$ to $100$ are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose $10 $ of these $100$ numbers. Then 10 of the integers from $1$ to $100$ are drawn, and a winning ticket is one which does not contain any of them. Prove that (a) if you buy $13$ tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket; (b) if you buy only $12$ tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers. (S Tokarev)

Two circles $ \left(\Omega_1\right),\left(\Omega_2\right) $ touch internally on the point $ T $. Let $ M,N $ be two points on the circle $ \left(\Omega_1\right) $ which are different from $ T $ and $ A,B,C,D $ be four points on $ \left(\Omega_2\right) $ such that the chords $ AB, CD $ pass through $ M,N $, respectively. Prove that if $ AC,BD,MN $ have a common point $ K $, then $ TK $ is the angle bisector of $ \angle MTN $. * $ \left(\Omega_2\right) $ is bigger than $ \left(\Omega_1\right) $

The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.

Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.

Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied: [list] [*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$; [*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$. [/list] A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.

Jose and Maria play the following game: Maria writes $2019$ positive integers different on the blackboard. Jose deletes some of them (possibly none, but not all) and write to the left of each of the remaining numbers a sign $+$or a sign $-$. Then the sum written on the board is calculated. If the result is a multiple of $2019$, Jose wins the game, if not, Maria wins. Determine which of the two has a winning strategy.

Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution: \[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]

Let $p$ be an odd prime and $S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}$, where $q = \frac{3p-5}{2}$. We write $\frac{1}{2}-2S_{q}$ in the form $\frac{m}{n}$, where $m$ and $n$ are integers. Prove that $m \equiv n (mod p)$

Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.

[u]Round 5[/u] [b]p13.[/b] The expression $\sqrt2 \times \sqrt[3]{3} \times \sqrt[6]{6}$ can be expressed as a single radical in the form $\sqrt[n]{m}$, where $m$ and $n$ are integers, and $n$ is as small as possible. What is the value of $m + n$? [b]p14.[/b] Bertie Bott also produces Bertie Bott’s Every Flavor Pez. Each package contains $6$ peppermint-, $2$ kumquat-, $3$ chili pepper-, and $5$ garlic-flavored candies in a random order. Harold opens a package and slips it into his Dumbledore-shaped Pez dispenser. What is the probability that of the first four candies, at least three are garlic-flavored? [b]p15.[/b] Quadrilateral $ABCD$ with $AB = BC = 1$ and $CD = DA = 2$ is circumscribed around and inscribed in two different circles. What is the area of the region between these circles? [u] Round 6[/u] [b]p16.[/b] Find all values of x that satisfy $\sqrt[3]{x^7} + \sqrt[3]{x^4} = \sqrt[3]{x}$. [b]p17.[/b] An octagon has vertices at $(2, 1)$, $(1, 2)$, $(-1, 2)$, $(-2, 1)$, $(-2, -1)$, $(-1, -2)$, $(1, -2)$, and $(2, -1)$. What is the minimum total area that must be cut off of the octagon so that the remaining figure is a regular octagon? [b]p18.[/b] Ron writes a $4$ digit number with no zeros. He tells Ronny that when he sums up all the two-digit numbers that are made by taking 2 consecutive digits of the number, he gets 99. He also reveals that his number is divisible by 8. What is the smallest possible number Ron could have written? [u]Round 7[/u] [b]p19.[/b] In a certain summer school, 30 kids enjoy geometry, 40 kids enjoy number theory, and 50 kids enjoy algebra. Also, the number of kids who only enjoy geometry is equal to the number of kids who only enjoy number theory and also equal to the number of kids who only enjoy algebra. What is the difference between the maximum and minimum possible numbers of kids who only enjoy geometry and algebra? [b]p20.[/b] A mouse is trying to run from the origin to a piece of cheese, located at $(4, 6)$, by traveling the shortest path possible along the lattice grid. However, on a lattice point within the region $\{0 \le x \le 4, 0 \le y \le 6$, $(x, y) \ne (0, 0),(4, 6)\}$ lies a rock through which the mouse cannot travel. The number of paths from which the mouse can choose depends on where the rock is placed. What is the difference between the maximum possible number of paths and the minimum possible number of paths available to the mouse? [b]p21.[/b] The nine points $(x, y)$ with $x, y \in \{-1, 0, 1\}$ are connected with horizontal and vertical segments to their nearest neighbors. Vikas starts at $(0, 1)$ and must travel to $(1, 0)$, $(0, -1)$, and $(-1, 0)$ in any order before returning to $(0, 1)$. However, he cannot travel to the origin $4$ times. If he wishes to travel the least distance possible throughout his journey, then how many possible paths can he take? [u]Round 8[/u] [b]p22.[/b] Let $g(x) = x^3 - x^2- 5x + 2$. If a, b, and c are the roots of g(x), then find the value of $((a + b)(b + c)(c + a))^3$. [b]p23.[/b] A regular octahedron composed of equilateral triangles of side length $1$ is contained within a larger tetrahedron such that the four faces of the tetrahedron coincide with four of the octahedron’s faces, none of which share an edge. What is the ratio of the volume of the octahedron to the volume of the tetrahedron? [b]p24.[/b] You are the lone soul at the south-west corner of a square within Elysium. Every turn, you have a $\frac13$ chance of remaining at your corner and a $\frac13$ chance of moving to each of the two closest corners. What is the probability that after four turns, you will have visited every corner at least once? PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Lavi Dopes has two boards $n \times n$. On the first board, he writes an integer in each of his $n^2$ squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if $n = 3$ and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this. Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board. (a) If $n = 4$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board? (b) If $n = 5$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?

How many ways are there to color the vertices of a square with $n$ colors $1,2, .. ., n$. (The colorings must be different so that we can’t get one from the other by a rotation.)

Determine all the integers solutions $(x,y)$ of the following equation $$\frac{x^2-4}{2x-1}+\frac{y^2-4}{2y-1}=x+y$$

Ana writes an infinite list of numbers using the following procedure. The first number of the list is a positive integer $a$ chosen by Ana. From there, each number in the list is obtained by calculating the sum of all the integers from $1$ to the last number written. For example, if $a = 3$, Ana's list starts as $3, 6, 21, 231, \dots$ because $1 + 2 + 3 = 6$, $1 + 2 + 3 + 4 + 5 + 6 = 21$ and $1 + 2 + 3 + \dots + 21 = 231$. Is it possible for all the numbers in Ana's list to be even?

Find, with proof, all functions $f$ mapping integers to integers with the property that for all integers $m,n$, $$f(m)+f(n)= \max\left(f(m+n),f(m-n)\right).$$