Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be [i]flooded[/i]. Vulcan is building a [i]levee[/i], which is a subset of unit edges of the grid (called [i]walls[/i]) forming a connected, non-self-intersecting path or loop*. The game then begins with Vulcan moving first. On each of Vulcan’s turns, he may add up to three new walls to the levee (maintaining the conditions for the levee). On each of Neptune’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Prove that Vulcan can always, in a finite number of turns, build the levee into a closed loop such that all flooded cells are contained in the interior of the loop, regardless of which cells Neptune initially floods. ----- [size=75]*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Vulcan cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.[/size]

Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$

Let $H$ be the orthocenter of triangle $ABC$, and $AD$, $BE$, $CF$ be the three altitudes of triangle $ABC$. Let $G$ be the orthogonal projection of $D$ onto $EF$, and $DD'$ be the diameter of the circumcircle of triangle $DEF$. Line $AG$ and the circumcircle of triangle $ABC$ intersect again at point $X$. Let $Y$ be the intersection of $GD'$ and $BC$, while $Z$ be the intersection of $AD'$ and $GH$. Prove that $X$, $Y$, and $Z$ are collinear. [i]Proposed by Li4 and Untro368.[/i]

Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$.

What is the maximal number of elements we can choose form the set $\{1, 2, \ldots, 31\}$, such that the sum of any two of them is not a perfect square.

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam. [hide=original wording]En un examen fueron propuestos 6 problemas. Cada problema fue resuelto por exactamente 1000 estudiantes, pero en ningun caso ha ocurrido que dos estudiantes en conjunto, hayan resuelto los 6 problemas. Determinar el menor numero de participantes que pudo haber en dicho exame[/hide]

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.

Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at points $M$ and $N$. The radii of $\omega_1$ and $\omega_2$ are $12$ and $15$, respectively, and $O_1O_2 = 18$. A point $X$ is chosen on segment $MN$. Line $O_1X$ intersects $\omega_2$ at points $A$ and $C$, where $A$ is inside $\omega_1$. Similarly, line $O_2X$ intersects $\omega_1$ at points $B$ and $D$, where $B$ is inside $\omega_2$. The perpendicular bisectors of segments $AB$ and $CD$ intersect at point $P$. Given that $PO_1 = 30$, find $PO_2^2$. [i]Proposed by Andrew Zhao[/i]

Lengths of two sides of a rectangle are $\sqrt2,1$. The rectangle rotates a round around one of its diagonal. Find the volume of the revolved body.

Henry places some rooks and some kings in distinct cells of a $2\times 8$ grid such that no two rooks attack each other and no two kings attack each other. Find the maximum possible number of pieces on the board. (Two rooks [i]attack[/i] each other if they are in the same row or column and no pieces are between them. Two kings attack each other if their cells share a vertex.)

Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that \[ (p(x)^2)'' \leq K(p'(x))^2. \]

Reals $a,b,c,x,y$ satisfy $a^2+b^2+c^2=x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$

Find the number of $10$ digit palindromes that are not divisible by $11$. [i]Lightning 1.3[/i]

For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd. [i]Proposed by Navid Safaie[/i]

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism? [i]Proposed by Deyuan Li and Andrew Milas[/i]

Point $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\cdots A_n,$ and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_n, A_1,$ and $B$ are consecutive vertices of a regular polygon?

Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a [i]follower[/i] of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$? [asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,S); label("$D$",D,N); label("$E$",E,S); label("$5$",A/2,W); label("$6$",(A+D)/2,N); [/asy] $\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$

Triangle $ABC$ is given. The median $CM$ intersects the circumference of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.

Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.

Let $AC$ be the largest side of the triangle $ABC$. The point M is selected on the ray $AC$ ray, and point $N$ on ray $CA$ such that $CN = CB$ and$ AM = AB$ . a) Prove that $\vartriangle ABC$ is isosceles if we know that $BM = BN$. b) Will the statement remain true if $AC$ is not necessarily the largest side of triangle $ABC$?

Let $ABC$ be a triangle, $H_a, H_b$ and $H_c$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_a, T_b, T_c$ its touchpoints of the incircle with the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $AH_bH_c$ and $AT_bT_c$ intersect again at $A'$. The circumcircles of triangles $BH_cH_a$ and $BT_cT_a$ intersect again at $B'$. The circumcircles of triangles $CH_aH_b$ and $CT_aT_b$ intersect again at $C'$. Prove that the points $A',B',C'$ are collinear. Malik Talbi

Let $J$ , $E$, $R$, and $Y$ be four positive integers chosen independently and uniformly at random from the set of factors of $1428$. What is the probability that $JERRY = 1428$? Express your answer in the form $\frac{a}{b\cdot 2^n}$ where $n$ is a nonnegative integer, $a $and $b$ are odd, and gcd $(a,b) = 1$.