On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.

A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.) (A. Shapovalov )

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X, Y,$ and $Z$ the midpoints of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$? $\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$

Given a triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove the theorem: In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.

The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$. Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel. [i]Advaith Avadhanam[/i]

[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.

$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.

A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon. Egor Bakaev

In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and $AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$. Prove: $\angle ACM_c = \angle KCB$.

In the figure, the triangles $ABC$ and $CDE$ are equilateral, with side lengths $1$ and $4$, respectively. Moreover, $B$, $C$ and $D$ are collinear and $F$ and $G$ are midpoints of $BC$ and $CD$, respectively. Let $P$ be the intersection point of $AF$ and $BE$. Determine the area of the shaded triangle $BPG$. [img]https://fv5-4.failiem.lv/thumb_show.php?i=qmpfykxcek&view&v=1&PHPSESSID=1f433228a75b4117c35f707722c547c423d3d671[/img]

Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r_2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.