In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

In pentagon $ABCDE$, $BE$ intersects $AC$ and $AD$ at $F$ and $G$, respectively. Suppose that $A[\vartriangle AF G] = A[\vartriangle BCF] = A[\vartriangle DEG] = 16$, where$ A[\vartriangle AF G]$ denotes the area of $\vartriangle AF G$. Next, suppose that $BF = 4$, $F G = 5$, and $GE = 6$. Compute $A[ABCDE]$.

If the circumradius of any three consecutive vertices of a convex polygon is at most $ 1, $ show that the discs of radius $ 1 $ centered at each vertex cover the polygon and its interior.

Let $S$ be a point inside $\angle pOq$, and let $k$ be a circle which contains $S$ and touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$ be the intersection point of the ray $PS$ and circumscribed circle of $\vartriangle SQR$ and $T \ne S$. Prove that $OT // SQ$ and $OT$ is a tangent of the circumscribed circle of $\vartriangle SQR$.

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people. P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$. [hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide] P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer $n$. ii. The first term of the sequence $(U_1)$ is $9n$. iii. For $k \geq 2$, $U_k = U_{k-1} - 17$. Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$. As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$. P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$. [url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed. P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.

At each vertex of a $4\times5$ rectangle there is a house. Find the path of the minimum length connecting all these houses.

Consider a curve with the following property: [i]inside the curve one can move an inscribed equilateral triangle so that each vertex of the triangle moves along the curve and draws the whole curve[/i]. Clearly, every circle possesses the property. Find a closed planar curve without self-intersections, that has the property but is not a circle.

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$

Find the least integer $n$ with the following property: For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points.

Given is triangle $ABC$ with $AB > AC$. Circles $O_B$, $O_C$ are inscribed in angle $BAC$ with $O_B$ tangent to $AB$ at $B$ and $O_C$ tangent to $AC$ at $C$. Tangent to $O_B$ from $C$ different than $AC$ intersects $AB$ at $K$ and tangent to $O_C$ from $B$ different than $AB$ intersects $AC$ at $L$. Line $KL$ and the angle bisector of $BAC$ intersect $BC$ at points $P$ and $M$, respectively. Prove that $BP = CM$.

Initially, three non-collinear points, $A$, $B$, and $C$, are marked on the plane. You have a pencil and a double-edged ruler of width $1$. Using them, you may perform the following operations: [list] [*]Mark an arbitrary point in the plane. [*]Mark an arbitrary point on an already drawn line. [*]If two points $P_1$ and $P_2$ are marked, draw the line connecting $P_1$ and $P_2$. [*]If two non-parallel lines $l_1$ and $l_2$ are drawn, mark the intersection of $l_1$ and $l_2$. [*]If a line $l$ is drawn, draw a line parallel to $l$ that is at distance $1$ away from $l$ (note that two such lines may be drawn). [/list] Prove that it is possible to mark the orthocenter of $ABC$ using these operations.

Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that (a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$. (b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$. ( V . V . Proizvolov , Moscow)

Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.

Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$ $ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$

Let $ABC$ be an acute triangle with $AB>AC$. The internal angle bisector of $\angle BAC$ meets $BC$ at $D$. Let $O$ be the circumcenter of $ABC$ and let $AO$ meet $BC$ at $E$. Let $J$ be the incenter of triangle $AED$. Show that if $\angle ADO=45^{\circ}$, then $OJ=JD$.

Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

Let $MN$ be a chord of the circle $\Gamma$ and let $S$ be the midpoint of $MN$. Let $A, B, C, D$ be points on $\Gamma$ such that $AC$ and $BD$ intersect at $S$ and $A$ and $B$ are on the same side of $MN$. Let $d_A, d_B, d_C , d_D$ be the distances from $MN$ to $A, B, C,$ and $D,$ respectively. Prove that $\frac{1}{d_A}+\frac{1}{d_D}=\frac{1}{d_B}+\frac{1}{d_C}$.

In convex quadrilateral $ ABCD$, $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, $ DA\equal{}d$, $ AC\equal{}e$, $ BD\equal{}f$. If $ \max \{a,b,c,d,e,f \}\equal{}1$, then find the maximum value of $ abcd$.

[u]Round 1[/u] [b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different? [img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img] [b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? [b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? [img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img] [b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img] [u]Round 2[/u] [b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$. [b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$. Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$ and perpendicular to $BC$. Let $M$ be a point on $BC$ such that $\angle AMB = \angle DMC$. If $AB = 3$, $BC = 24$, and $CD = 4$, what is the value of $AM + MD$?

Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.