Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $L=\int_{-\infty}^{\infty} f(x) dx$ exists. Show that $$\int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx=L.$$

The 20 pupils in a class each send 10 cards to 10 (different) class members. [size=92][i][note: you cannot send a card to yourself.][/i][/size] (a) Show at least 2 pupils sent each other a card. (b) Now suppose we had $n$ pupils sending $m$ cards each. For which $(m,n)$ is the above true? (That is, find minimal $m(n)$ or maximal $n(m)$)

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$. Prove that $M$, $N$, and $P$ lie on a line.

A conference is attended by $n (n\ge 3)$ scientists. Each scientist has some friends in this conference (friendship is mutual and no one is a friend of him/herself). Suppose that no matter how we partition the scientists into two nonempty groups, there always exist two scientists in the same group who are friends, and there always exist two scientists in different groups who are friends. A proposal is introduced on the first day of the conference. Each of the scientists' opinion on the proposal can be expressed as a non-negative integer. Everyday from the second day onwards, each scientists' opinion is changed to the integer part of the average of his/her friends' opinions from the previous day. Prove that after a period of time, all scientists have the same opinion on the proposal.

Find all functions $f: \mathbb{R}\to [0;+\infty)$ such that: \[f(x^2+y^2)=f(x^2-y^2)+f(2xy)\] for all real numbers $x$ and $y$. [i]Laurentiu Panaitopol[/i]

Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$. [i]Proposed by Carl Schildkraut[/i]

Let the medians of the triangle $ABC$ intersect at point $M$. A line $t$ through $M$ intersects the circumcircle of $ABC$ at $X$ and $Y$ so that $A$ and $C$ lie on the same side of $t$. Prove that $BX\cdot BY=AX\cdot AY+CX\cdot CY$.

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 \( f(2023) \).