Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent. $\text{Vangelis Psychas and Silouanos Brazitikos}$

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.

Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed?

Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order. Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal. [i]Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane[/i]

The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$. (Gregory Filippovsky)

Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.

Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$. [i]Proposed by Taco12[/i]

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to [asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75)); [/asy] $ \textbf{(A)}\ k \qquad\textbf{(B)}\ \frac{6k}{5} \qquad\textbf{(C)}\ \frac{9k}{8} \qquad\textbf{(D)}\ \frac{5k}{4} \qquad\textbf{(E)}\ 2k $

The diameter \( AD \) of the circumcircle of triangle \( ABC \) intersects line \( BC \) at point \( K \). Point \( D \) is reflected symmetrically with respect to point \( K \), resulting in point \( L \). On line \( AB \), a point \( F \) is chosen such that \( FL \perp AC \). Prove that \( FK \perp AD \). [i]Proposed by Matthew Kurskyi[/i]

From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.

Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?

Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.

Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$, then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$”. Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$, but none with weight $0$.

The circle k intersects the sides $BC$, $CA$, $AB$ of triangle $ABC$ in points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$. The perpendiculars to $BC$, $CA$, $AB$ through $A_1$, $B_1$, $C_1$, respectively, meet at a point $M$. Prove that the three perpendiculars to $BC$, $CA$, $AB$ through $A_2$, $B_2$, and $C_2$, respectively, also meet in one point.

$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

[b]p13.[/b] Suppose $\overline{AB}$ is a radius of a circle. If a point $C$ is chosen uniformly at random inside the circle, what is the probability that triangle $ABC$ has an obtuse angle? [b]p14.[/b] Find the second smallest positive integer $c$ such that there exist positive integers $a$ and $b$ satisfying the following conditions: $\bullet$ $5a = b = \frac{c}{5} + 6$. $\bullet$ $a + b + c$ is a perfect square. [b]p15.[/b] A spotted lanternfly is at point $(0, 0, 0)$, and it wants to land on an unassuming CMU student at point $(2, 3, 4)$. It can move one unit at a time in either the $+x$, $+y$, or $+z$ directions. However, there is another student waiting at $(1, 2, 3)$ who will stomp on the lanternfly if it passes through that point. How many paths can the lanternfly take to reach its target without getting stomped? PS. You should use hide for answers.