Let $ABC$ be an acute triangle such that $AB<AC$ with orthocenter $H$. The altitudes $BH$ and $CH$ intersect $AC$ and $AB$ at $B_{1}$ and $C_{1}$. Denote by $M$ the midpoint of $BC$. Let $l$ be the line parallel to $BC$ passing through $A$. The circle around $ CMC_{1}$ meets the line $l$ at points $X$ and $Y$, such that $X$ is on the same side of the line $AH$ as $B$ and $Y$ is on the same side of $AH$ as $C$. The lines $MX$ and $MY$ intersect $CC_{1}$ at $U$ and $V$ respectively. Show that the circumcircles of $ MUV$ and $ B_{1}C_{1}H$ are tangent. [i] Authored by Nikola Velov[/i]

The center of the inscribed circle of triangle $ABC$ is $I$. Let $e$ be the perpendicular line on $CI$ passing through $I$. The line $e$ itnersects the side $AC$ at $A'$ and the side $BC$ at point $B'$. Let $A''$ be the symmetric point of $A$ wrt $A'$, $B''$ be the symmetric point of $B$ wrt $B'$. Prove that $A''B''$ is a line tangent to the incircle.

We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$? b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

[b]p1.[/b] In the correct identity $(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)$ two numbers were replaced with dots. What were these numbers? [b]p2.[/b] A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? [img]https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png[/img] [b]p3.[/b] Find the angle between the hour and minute hands at $7$ hours $38$ minutes. [b]p4.[/b] The lottery game is played as follows. A random number from $1$ to $1000$ is selected. If it is divisible by $2$, they pay a ruble, if it is divisible by $10$ - two rubles, by $12$ - four rubles, by $20$ - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options. [b]p5.[/b]The sum of the digits of a positive integer $x$ is equal to $n$. Prove that between $x$ and $10x$ you can find an integer whose sum of digits is $ n + 5$. [b]p6.[/b] $9$ people took part in the campaign, which lasted $12$ days. There were $3$ people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all $12$ days they were able to appoint three people on duty, taking into account this requirement. Could this be so? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Given a circle and two tangents to this circle. Draw a third tangent to the circle in such a way that its segment contained by the given tangents has the given length $ d $.

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.

[b]p1.[/b] Consider a $4 \times 4$ lattice on the coordinate plane. At $(0,0)$ is Mori’s house, and at $(4,4)$ is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at $(2, 2)$ was closed. How many ways are there now for Mori to go to work? [b]p2.[/b] Given two integers, define an operation $*$ such that if a and b are integers, then a $*$ b is an integer. The operation $*$ has the following properties: 1. $a * a$ = 0 for all integers $a$. 2. $(ka + b) * a = b * a$ for integers $a, b, k$. 3. $0 \le b * a < a$. 4. If $0 \le b < a$, then $b * a = b$. Find $2017 * 16$. [b]p3.[/b] Let $ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $CA = 15$. Let $A'$, $B'$, $C'$, be the midpoints of $BC$, $CA$, and $AB$, respectively. What is the ratio of the area of triangle $ABC$ to the area of triangle $A'B'C'$? [b]p4.[/b] In a strange world, each orange has a label, a number from $0$ to $10$ inclusive, and there are an infinite number of oranges of each label. Oranges with the same label are considered indistinguishable. Sally has 3 boxes, and randomly puts oranges in her boxes such that (a) If she puts an orange labelled a in a box (where a is any number from 0 to 10), she cannot put any other oranges labelled a in that box. (b) If any two boxes contain an orange that have the same labelling, the third box must also contain an orange with that labelling. (c) The three boxes collectively contain all types of oranges (oranges of any label). The number of possible ways Sally can put oranges in her $3$ boxes is $N$, which can be written as the product of primes: $$p_1^{e_1} p_2^{e_2}... p_k^{e_k}$$ where $p_1 \ne p_2 \ne p_3 ... \ne p_k$ and $p_i$ are all primes and $e_i$ are all positive integers. What is the sum $e_1 + e_2 + e_3 +...+ e_k$? [b]p5.[/b] Suppose I want to stack $2017$ identical boxes. After placing the first box, every subsequent box must either be placed on top of another one or begin a new stack to the right of the rightmost pile. How many different ways can I stack the boxes, if the order I stack them doesn’t matter? Express your answer as $$p_1^{e_1} p_2^{e_2}... p_n^{e_n}$$ where $p_1, p_2, p_3, ... , p_n$ are distinct primes and $e_i$ are all positive integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a figure $B$ in the plane, consider the figures $A$, containing $\mathcal B$, with the property [i]$(P)$: a composition of an odd number of central symmetries with centers in $A$ is also a central symmetry with center in $A$.[/i] The task of this problem is to determine the smallest such figure, denoted by $\mathcal A$, that is contained in every figure $A$. (a) Determine the figure $\mathcal A$ if $B$ consists of: $(1)$ two distinct points $I,J$; $(2)$ three non-collinear points $I,J,K$. (b) Determine $\mathcal A$ if $B$ is a circle (with nonzero radius). (c) Give some examples of figures $B$ whose associated figures $\mathcal A$ are mutually distinct and distinct from the above ones.

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with, $AB$ ≠ $AC$, and let $K$ is your incenter. The points $P$ and $Q$ are the points of the intersections of the circumcicle($BCK$) with the line(s) $AB$ and $AC$, respectively. Let $D$ be intersection of $AK$ and $BC$. Show that $P, Q, D$ are collinears.

A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-2.687,0),E); label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$. [i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. [i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then \[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\] where $r_1$ is the exradius corresponding to the vertex $A.$

Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$ respectively Let $M$ and $N$ be symmetric to $B$ and $C$ about $B_1$ and $C_1$ respectively. Prove that the lines $KM$ and $LN$ meet on $BC$.

The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$..

Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt 5$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt 5$. What is $|AD|$? $ \textbf{(A)}\ 10\sqrt 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 4\sqrt{15} \qquad\textbf{(D)}\ 7\sqrt 5 \qquad\textbf{(E)}\ \text{None of above} $

Given a quadrilateral \(ABCD\), point \(M\) is the midpoint of the side \(CD\). It turns out that \(\angle BMA = 90^{\circ}\) and \(\angle MAB = \angle CBD\). Prove that \(AC = AB\). [i]Proposed by Anton Trygub[/i]

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

Given a triangle $ABC$, let $D$ be a point on segment $BC$. Construct the circumcircle $\omega$ of triangle $ABD$ and point $E$ on $\omega$ such that $CE$ is tangent to $\omega$ and $A, E$ are on opposite sides of $BC$ (as shown in diagram). If $\angle CAD = \angle ECD$ and $AC = 12$, $AB = 7$, find $AE$.