Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear. [i]Proposed by Dominik Burek, Poland[/i]

Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

Let a line, perpendicular to side $AD$ of parallelogram $ABCD$ passing through point $B$, intersect line $CD$ at point $M$, and a line, passing through point $B$ and perpendicular to side $CD$, intersect line $AD$ at point $N$. Prove that the line passing through point $B$ perpendicular to the diagonal $AC$, passes through the midpoint of the segment $MN$.

In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \geq MA$.

Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be random integers chosen independently and uniformly from the set $\{ 0, 1, 2, \dots, 23 \}$. (Note that the integers are not necessarily distinct.) Find the probability that \[ \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0. \] (Here $\operatorname{cis} \theta$ means $\cos \theta + i \sin \theta$.)

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$? [asy] size(140); pair A, B, C, D, E, F, X, Y; real length = 12.5; real width = 10; A = origin; B = (length, 0); C = (length, width); D = (0, width); X = rotate(330, C)*B; E = extension(C, X, A, B); Y = rotate(30, C)*D; F = extension(C, Y, A, D); draw(E--C--F); label("$2$", A--F, dir(180)); label("$6$", E--B, dir(270)); draw(A--B--C--D--cycle); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$", A, dir(225)); label("$B$", B, dir(315)); label("$C$", C, dir(45)); label("$D$", D, dir(135)); label("$E$", E, dir(270)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$

Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions: [list] [*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$. [*]For any prime number $p$ and for any index $n \geqslant 1$, the number \[ a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p} \] is a multiple of $p$. [/list]

Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

Let $ABCD$ be a square with side $10$. Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$, one with midpoint $M$ and radius $|AM|$, and one with midpoint $N$ and radius $|BN|$. The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$.

Prove that for any integer $a \ge 2$ and positive integer $n,$ there exist positive integer $k$ such that $a^k+1,a^k+2,\ldots,a^k+n$ are all composite numbers.

Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds. $$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$ [i]Costică Ambrinoc[/i]

Compute $$\int\frac{(\sin x+\cos x)(4-2\sin2x-\sin^22x)e^x}{\sin^32x}dx$$ where $x\in\left(0,\frac\pi2\right)$. [i]Proposed by Mihály Bencze[/i]

Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$. Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$. [i]A. Kovaldji, V. Senderov[/i]

In a soccer championship $2004$ teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows: a) any two teams can play against one each other at most one game; b) from any $4$ teams, $3$ of them play against one each other. How many days are necessary to make such a championship, knowing that each team can play at most one game per day?

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$ [i]Proposed by Dusan Djukic $IMO \ Shortlist \ 2013$[/i]

Let $p$ of prime of the form $3k+2$ such that $a^2+ab+b^2$ is divisible by $p$ for some integers $a$ and $b$. Prove that both of $a$ and $b$ are divisible by $p$.

There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$. (R Zhenodarov)

Let $a, b, c, d$ be four real numbers such that \begin{align*}a + b + c + d &= 8, \\ ab + ac + ad + bc + bd + cd &= 12.\end{align*} Find the greatest possible value of $d$.