In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation. (1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$. (2) Find the volume of the common part of $V_1$ and $V_2$.

In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric. [i]Proposed by Iman Maghsoudi - Iran[/i]

Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$. $M$- midpoint of $AC$. Prove, that $AE,CF,DM$ intersects at one point.

Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

The polynomial $ (x\plus{}y)^9$ is expanded in decreasing powers of $ x$. The second and third terms have equal values when evaluated at $ x\equal{}p$ and $ y\equal{}q$, where $ p$ and $ q$ are positive numbers whose sum is one. What is the value of $ p$? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9$

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties broken arbitrarily). Over all possible sets of orders, what is the maximum total amount the university could have paid? [i]Proposed by Evan Chen[/i]

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? [asy]draw((0,0)--(7,0)); draw((0,0)--(33/7,7.66651)); draw((33/7,7.66651)--(7,0)); draw((11/5,7*7.66651/15)--(7,0)); draw((63/17,0)--(33/7,7.66651)); draw((0,0)--(45/7,7.66651/4)); dot((0,0)); label("A",(0,0),SW); dot((7,0)); label("B",(7,0),SE); dot((33/7,7.66651)); label("C",(33/7,7.66651),N); dot((11/5,7*7.66651/15)); label("D",(11/5+.2,7*7.66651/15-.25),S); dot((63/17,0)); label("E",(63/17,0),NE); dot((45/7,7.66651/4)); label("H",(44/7,7.66651/4),NW); dot((27/7,3*7.66651/20)); label("P",(27/7,3*7.66651/20),NW); dot((5,7*7.66651/36)); label("Q",(5,7*7.66651/36),N); label("9",(33/14,7.66651/2),NW); label("8",(41/7,7.66651/2),NE); label("7",(3.5,0),S);[/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad \textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad \textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad \textbf{(E)}\ \frac{6}{5}$

Given an acute triangle $ABC$ with altitudes $AA_1$, $BB_1$, and $CC_1$ ($A_1 \in BC$, $B_1 \in AC$, $C_1 \in AB$) and circumcircle $k$, the rays $B_1A_1$, $C_1B_1$, and $A_1C_1$ meet $k$ at points $A_2$, $B_2$, and $C_2$, respectively. Find the maximum possible value of \[ \sin \angle ABB_2 \cdot \sin \angle BCC_2 \cdot \sin \angle CAA_2 \] and all acute triangles $ABC$ for which it is achieved.

In a forest each of $n$ animals ($n\ge 3$) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths for moving from cave to cave, never turns from one path to another between the caves and returns to its own cave in the end of its campaign. It is also known that no path between two caves is used by more than one campaign-making animal. a) Prove that for any prime $n$, the maximum possible number of campaign-making animals is $\frac{n-1}{2}$. b) Find the maximum number of campaign-making animals for $n=9$.

The sum of the digits in base ten of $ (10^{4n^2\plus{}8}\plus{}1)^2$, where $ n$ is a positive integer, is $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4n \qquad \textbf{(C)}\ 2\plus{}2n \qquad \textbf{(D)}\ 4n^2 \qquad \textbf{(E)}\ n^2\plus{}n\plus{}2$

Consider the closed plane curves $C_i$ and $C_o,$ their respective lengths $|C_i|$ and $|C_o|,$ the closed surfaces $S_i$ and $S_o,$ and their respective areas $|S_i|$ and $|S_o|.$ Assume that $C_i$ lies inside $C_o$ and $S_i$ inside $S_o.$ (Subscript $i$ stands for "inner," $o$ for "outer.") Prove the correct assertions among the following four, and disprove the others. (i) If $C_i$ is convex, $|C_i| \le |C_o|.$ (ii) If $S_i$ is convex, $|S_i| \le |S_o|.$ (iii) If $C_o$ is the smallest convex curve containing $C_i,$ then $|C_o| \le |C_i|.$ (iv) If $S_o$ is the smallest convex surface containing $S_i,$ then $|S_o| \le |S_i|.$ You may assume that $C_i$ and $C_o$ are polygons and $S_i$ and $S_o$ polyhedra.

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

An integer sequence satisfies $a_{n+1}={a_n}^3 +1999$. Show that it contains at most one square.

$4.$ How many line segments have both their endpoints located at the vertices of a given cube $?$

Consider a function \( n \mapsto f(n) \) that satisfies the following conditions: \( f(n) \) is an integer for each \( n \). \( f(0) = 1 \). \( f(n+1) > f(n) + f(n-1) + \cdots + f(0) \) for each \( n = 0, 1, 2, \dots \). Determine the smallest possible value of \( f(2023) \).

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.

Nick has a $3\times3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.

Find the graph of the function $y=x-|x+x^2|$

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

Let $ABCDEF$ be a regular hexagon. Let $P$ be the intersection point of $\overline{AC}$ and $\overline{BD}$. Suppose that the area of triangle $EFP$ is 25. What is the area of the hexagon?

In a right triangle $ABC$ with a right angle $C$, let $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square? $\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$