In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does. On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed. Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game? [i]Proposed by Andrew Gu[/i]

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?

You are given a ruler with two parallel straight edges a distance $d$ apart. It may be used (1) to draw the line through two points, (2) given two points a distance $\ge d$ apart, to draw two parallel lines, one through each point, (3) to draw a line parallel to a given line, a distance d away. One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line. Show how to construct : (A) the bisector of a given angle, and (B) the perpendicular to the midpoint of a given line segment.

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\] where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with [list] [*] $a_1=c$, and [*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ [/list] Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? [i]Proposed by chorn[/i]

The circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, point $M$ is the midpoint of $AB$. On line $AB$ select points $S_1$ and $S_2$. Let $S_1X_1$ and $S_1Y_1$ be tangents drawn from $S_1$ to circle $\omega_1$, similarly $S_2X_2$ and $S_2Y_2$ are tangents drawn from $S_2$ to circles $\omega_2$. Prove that if the point $M$ lies on the line $X_1X_2$, then it also lies on the line $Y_1Y_2$.

Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.

Let $S$ be the set of all ordered pairs of integers $(m,n)$ satisfying $m>0$ and $n<0.$ Let $<$ be a partial ordering on $S$ defined by the statement $(m,n)<(m',n')$ if and only if $m\le m'$ and $n\le n'.$ An example is $(5,-10)<(8,-2).$ Now let $O$ be a completely ordered subset of $S,$ in other words if $(a,b)\in O$ and $(c,d) \in O,$ then $(a,b)<(c,d)$ or $(c,d)<(a,b).$ Also let $O'$ denote the collection of all such completely ordered sets. (a) Determine whether and arbitrary $O\in O'$ is finite. (b) Determine whether the carnality $|O|$ of $O$ is bounded for $O\in O'.$ (c) Determine whether $|O|$ can be countable infinite for any $O\in O'.$

Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.) (Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)

Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$. The set $M$ is completed with new fractions according to the rule: take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc. Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.

Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ -14 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find all nonnegative integers $x, y, z$ satisfying the equation $$2^x+31^y=z^2.$$

Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$. Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$. (Note. The smaller $C$, the better the solution.)

Consider a homogeneous function with degree $4$. $f(x,y,z)=a_1x^4+a_2y^4+a_3z^4+3a_4x^2y^2+3a_5y^2z^2+3a_6x^2z^2$. Find $\oiint_{\sum} f(x,y,z)dS$, where $\sum: x^2+y^2+z^2=1$.

Let the Sierpinski triangle graph $S_n$ be defined as follows. $S_0$ consists of three vertices and three edges in a triangle. $S_1$ consists of $6$ vertices and $9$ edges. To make $S_n+1,$ we take three copies of $S_n$ and merge vertices at the corners, where the bottom-left corner of the top copy merges with the top corner of the bottom-left copy, etc. Then the number of cycles on $S_4,$ which visit each vertex exactly once and traverse each edge at most once, can be expressed as $p_1^{e_1}p_2^{e_2}$ for some primes $p_1, p_2$ and positive integers $e_1, e_2.$ Find $p_1 + p_2 + e_1 + e_2.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/6/2/51d83da65910cd32ce0b235a9615ec467870e1.png[/img] [/center]

In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class?

You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations: 1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta. 2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up. 3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively. 4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair. 5. Remove four consecutive beads of one color. Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'', b) ``cyan, yellow, cyan, magenta, cyan'', c) ``magenta, magenta, cyan, cyan, cyan'', d) ``yellow, cyan, cyan, cyan''. [i]Proposed by Sammy Luo[/i]

Let $n$ be a positive integer and consider the system \begin{align*} S(n):\begin{cases} x^2+ny^2=z^2\\ nx^2+y^2=t^2 \end{cases}, \end{align*} where $x,y,z$, and $t$ are naturals. If [list] [*] $M_1=\{n\in\mathbb N:$ system $S(n)$ has infinitely many solutions$\}$, and [*] $M_1=\{n\in\mathbb N:$ system $S(n)$ has no solutions$\}$, [/list] prove that [list] [*] $7 \in M_1$ and $10 \in M_2$. [*] sets $M_1$ and $M_2$ are infinite. [/list]

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Basil drew several circles on the plane and drew all common tangent lines for all pairs of circles. It turned out that the lines contain all sides of a regular polygon with 2011 vertices. What is the smallest possible number of circles? (Author: N. Agahanov)

Find all integers $a,b,c,d$ such that (i) $1 \leq a \leq b \leq c \leq d$; (ii) $ab + cd = a +b +c +d + 3$.

We are given a line $ g$ with four successive points $ P$, $ Q$, $ R$, $ S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ ABCD$ such that $ P$ lies on the line $ AD$, $ Q$ on the line $ BC$, $ R$ on the line $ AB$ and $ S$ on the line $ CD$.

Let $ABC$ be a scalene acute triangle. Let $B_1$ be the point on ray $[AC$ such that $|AB_1|=|BB_1|$. Let $C_1$ be the point on ray $[AB$ such that $|AC_1|=|CC_1|$. Let $B_2$ and $C_2$ be the points on line $BC$ such that $|AB_2|=|CB_2|$ and $|BC_2|=|AC_2|$. Prove that $B_1$, $C_1$, $B_2$, $C_2$ are concyclic.

Let $ n$ be a natural number, and $ n \equal{} 2^{2007}k\plus{}1$, such that $ k$ is an odd number. Prove that \[ n\not|2^{n\minus{}1}\plus{}1\]

Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$, and let $K$ be the foot of the altitude from $Y$ to $AB$. Let $O$ denote the midpoint of $AB$ and $L$ be the intersection of $XZ$ with $YO$. Select a point $M$ on line $KL$ with $MA=MB$ , and finally, let $I$ be the reflection of $O$ across $XZ$. Prove that if quadrilateral $XKOZ$ is cyclic then so is quadrilateral $YOMI$. [i]Proposed by Evan Chen[/i]