Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

For some positive integer $n$, $n$ is considered a $\textbf{unique}$ number if for any other positive integer $m\neq n$, $\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\}$ holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.

Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$, (i) $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$. (ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.

Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

Consider the following sequence $$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$. (Proposed by Tomas Barta, Charles University, Prague)

Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?

Let $a$ and $b$ be nonnegative real numbers. Prove that \[\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right) \le 3(a^2+b^2).\]

Initially, the numbers $2$ and $5$ are written on the board. A \emph{move} consists of replacing one of the two numbers on the board with their sum. Is it possible to obtain (in a finite numer of moves) a situation in which the two integers written on the board are consecutive? Justify your answer.

Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

Compute the last digit of $(5^{20}+2)^3.$

Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.

Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that \[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]

A point object of mass $m$ is connected to a cylinder of radius $R$ via a massless rope. At time $t = 0$ the object is moving with an initial velocity $v_0$ perpendicular to the rope, the rope has a length $L_0$, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds $T_{max}$. Express your answers in terms of $T_{max}$, $m$, $L_0$, $R$, and $v_0$. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy] What is the angular momentum of the object with respect to the axis of the cylinder at the instant that the rope breaks? $ \textbf{(A)}\ mv_0R $ $ \textbf{(B)}\ \frac{m^2v_0^3}{T_{max}} $ $ \textbf{(C)}\ mv_0L_0 $ $ \textbf{(D)}\ \frac{T_{max}R^2}{v_0} $ $ \textbf{(E)}\ \text{none of the above} $

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

Let $S$ be a given sphere with center $O$ and radius $r$. Let $P$ be any point outside then sphere $S$, and let $S'$ be the sphere with center $P$ and radius $PO$. Denote by $F$ the area of the surface of the part of $S'$ that lies inside $S$. Prove that $F$ is independent of the particular point $P$ chosen.

To unlock his cell phone, Joao slides his finger horizontally or vertically across a numerical box, similar to the one represented in the figure, describing a $7$-digit code, without ever passing through the same digit twice. For example, to indicate the code $1452369$, Joao follows the path indicated in the figure. [img]https://cdn.artofproblemsolving.com/attachments/8/a/511018ba4e43c2c6f0be350d57161eb5ea7c2b.png[/img] João forgot his code, but he remembers that it is divisible by $9$. How many codes are there under these conditions?

Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$

Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. [i]Ray Li.[/i]

Let $E: R^n \backslash \{0\} \to R^+$ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and $p \in R^n \backslash \{0\}$ , $E (\lambda p) = \lambda^2 E (p)$). Prove that if the second derivative of $E''(p): R^n \times R^n \to R$ is a non-degenerate bilinear form at any point $p \in R^n \backslash \{0\}$, then $E''(p)$ ($p \in R^n \backslash \{0\}$) is positive definite.

Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.) [asy]unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp); [/asy] $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $