A convex quadrilateral $ABCD$ is inscribed in a circle $\omega$. The rays $AD$ and $BC$ meet in point $K$ and the rays $AB$ and $DC$ meet in $L$. Prove that the circumcircle of triangle $AKL$ is tangent to $\omega$ if and only if so is the circumcircle of triangle $CKL$.

Factor completely the expression $(a-b)^3+(b-c)^3+(c-a)^3$

A triangle $ABC$ is inscribed in the circle $(O)$, and has incircle $(I)$. The circles with diameter $IA$ meets $(O)$ at $A_1$ distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Line $B_1C_1$ meets $BC$ at $A_2$, and points $B_2,C_2$ are defined in the same manner. Prove that $O$ is the orthocenter of triangle $A_2B_2C_2$. Trần Minh Ngọc

Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Consider a triangle $ABC$ with height $AH$ and $H$ on $BC$. Let $\gamma_1$ and $\gamma_2$ be the circles with diameter $BH,CH$ respectively, and let their centers be $O_1$ and $O_2$. Points $X,Y$ lie on $\gamma_1,\gamma_2$ respectively such that $AX,AY$ are tangent to each circle and $X,Y,H$ are all distinct. $P$ is a point such that $PO_1$ is perpendicular to $BX$ and $PO_2$ is perpendicular to $CY$. Prove that the circumcircles of $PXY$ and $AO_1O_2$ are tangent to each other.

The area of polygon $ ABCDEF$, in square units, is [asy]draw((0,0)--(4,0)--(4,9)--(-2,9)--(-2,4)--(0,4)--cycle); label("A",(-2,9),NW); label("B",(4,9),NE); label("C",(4,0),SE); label("D",(0,0),SW); label("E",(0,4),NE); label("F",(-2,4),SW); label("5",(-2,6.5),W); label("4",(2,0),S); label("9",(4,4.5),E); label("6",(1,9),N); label("All angles in this diagram are right.",(0,-3),S);[/asy] \[ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74 \]

Regular $n$-gon is divided to triangles using $n-3$ diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent? [i]Proposed by Dusan Djukic[/i]

Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$. [i]Proposed by Evan Chen[/i]

Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$. On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$. Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$. [asy] unitsize(1.5 cm); pair[] A, B, C; A[0] = (0,0); B[0] = (1,0); C[0] = dir(60); A[1] = B[0] + dir(-60); A[2] = C[0] + dir(120); B[1] = C[0] + dir(60); B[2] = A[0] + dir(240); C[1] = A[0] + (-1,0); C[2] = B[0] + (1,0); draw(A[1]--A[2]); draw(B[1]--B[2]); draw(C[1]--C[2]); draw(circumcircle(A[2],B[1],C[2])); dot("$A$", A[0], SE); dot("$A_1$", A[1], SE); dot("$A_2$", A[2], NW); dot("$B$", B[0], SW); dot("$B_1$", B[1], NE); dot("$B_2$", B[2], SW); dot("$C$", C[0], N); dot("$C_1$", C[1], W); dot("$C_2$", C[2], E); [/asy]

In a right triangle, the point $O$ is the center of the circumcircle. Another circle of smaller radius centered at the point $O$ touches the larger leg and the altitude drawn from the top of the right angle. Find the acute angles of a right triangle and the ratio of the radii of the circumscribed and smaller circles.

The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers $m$, $n$, and $p$ such that the ratio of the area of the hexagon to the area of the square can be written as $\tfrac{m+\sqrt{n}}{p}$ where $m$ and $p$ are relatively prime. Find $m + n + p$. [asy] import graph; size(4cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); filldraw((1.13,2.5)--(-0.13,2.5)--(-0.13,1.23)--(1.13,1.23)--cycle,grey); draw((0,1)--(1,1)); draw((1,1)--(1.5,1.87)); draw((1.5,1.87)--(1,2.73)); draw((1,2.73)--(0,2.73)); draw((0,2.73)--(-0.5,1.87)); draw((-0.5,1.87)--(0,1)); draw((1.13,2.5)--(-0.13,2.5)); draw((-0.13,2.5)--(-0.13,1.23)); draw((-0.13,1.23)--(1.13,1.23)); draw((1.13,1.23)--(1.13,2.5)); [/asy]

Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.

Let $ABCD$ be a convex quadrilateral and $K$ be the common point of rays $AB$ and $DC$. There exists a point $P$ on the bisectrix of angle $AKD$ such that lines $BP$ and $CP$ bisect segments $AC$ and $BD$ respectively. Prove that $AB = CD$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Two circles intersect at points $A\neq B$. A line passing through $A{}$ intersects the circles again at $C$ and $D$. Let $E$ and $F$ be the midpoints of the arcs $\overarc{BC}$ and $\overarc{BD}$ which do not contain $A{}$ and let $M$ be the midpoint of the segment $CD$. Prove that $ME$ and $MF$ are perpendicular.

Find the triangle of the least area which can cover any triangle with sides not exceeding $1$.

A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point. (R. Krutovsky, A. Yakubov)

Let $O{}$ be the circumcenter of the triangle $ABC$. On the rays $AC$ and $BC$ consider the points $C_a$ and $C_b$ respectively, such that $AC_a$ and $BC_b$ are equal in length to $AB$. Let $O_c{}$ be the circumcenter of the triangle $CC_aC_b$. Define the points $O_a{}$ and $O_b{}$ similarly. Prove that $O{}$ is the orthocenter of the triangle $O_aO_bO_c$. [i]Proposed by A. Zaslavsky[/i]

In triangle $ABC$ with incentre $I$, let $M_A,M_B$ and $M_C$ by the midpoints of $BC, CA$ and $AB$ respectively, and $H_A,H_B$ and $H_C$ be the feet of the altitudes from $A,B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent tot he circumcircle of triangle $ABC$ and passing through $B$, and denote by $\ell_b'$ the reflection of $\ell_b$ in $BI$. Let $P_B$ by the intersection of $M_AM_C$ and $\ell_b$, and let $Q_B$ be the intersection of $H_AH_C$ and $\ell_b'$. Defined $\ell_c,\ell_c',P_C,Q_C$ analogously. If $R$ is the intersection of $P_BQ_B$ and $P_CQ_C$, prove that $RB=RC$.

We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

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 $\triangle XYZ$? $\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$

Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point. Trần Quang Hùng

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$. Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.

[b]p1.[/b] Catherine's teacher thinks of a number and asks her to subtract $5$ and then multiply the result by $6$. Catherine accidentally switches the numbers by subtracting 6 and multiplying by $5$ to get $30$. If Catherine had not swapped the numbers, what would the correct answer be? [b]p2.[/b] At Acton Boxborough Regional High School, desks are arranged in a rectangular grid-like configuration. In order to maintain proper social distancing, desks are required to be at least 6 feet away from all other desks. Assuming that the size of the desks is negligible, what is the maximum number of desks that can fit in a $25$ feet by $25$ feet classroom? [b]p3.[/b] Joshua hates writing essays for homework, but his teacher Mr. Meesh assigns two essays every $3$ weeks. However, Mr. Meesh favors Joshua, so he allows Joshua to skip one essay out of every $4$ that are assigned. How many essays does Joshua have to write in a $24$-week school year? [b]p4.[/b] Libra likes to read, but she is easily distracted. If a page number is even, she reads the page twice. If a page number is an odd multiple of three, she skips it. Otherwise, she reads the page exactly once. If Libra's book is $405$ pages long, how many pages in total does she read if she starts on page $1$? (Reading the same page twice counts as two pages.) [b]p5.[/b] Let the GDP of an integer be its Greatest Divisor that is Prime. For example, the GDP of $14$ is $7$. Find the largest integer less than $100$ that has a GDP of $3$. [b]p6.[/b] As has been proven by countless scientific papers, the Earth is a flat circle. Bob stands at a point on the Earth such that if he walks in a straight line, the maximum possible distance he can travel before he falls off is $7$ miles, and the minimum possible distance he can travel before he falls off is $3$ miles. Then the Earth's area in square miles is $k\pi$ for some integer $k$. Compute $k$. [b]p7.[/b] Edward has $2$ magical eggs. Every minute, each magical egg that Edward has will double itself. But there's a catch. At the end of every minute, Edward's brother Eliot will come outside and smash one egg on his forehead, causing Edward to lose that egg permanently. For example, starting with $2$ eggs, after one minute there will be $3$ eggs, then $5$, $9$, and so on. After $1$ hour, the number of eggs can be expressed as $a^b + c$ for positive integers $a$, $b$, $c$ where $a > 1$, and $a$ and $c$ are as small as possible. Find $a + b + c$. [b]p8.[/b] Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $..$, $a_{2019}$, $a_{2020}$ with the property that $a_n =\frac{a_{n-1} + a_n + a_{n+1}}{3}$ for all $n = 2$, $3$, $4$, $5$,$...$, $2018$, $2019$. Given that $a_1 = 1$ and $a_{1000} = 1999$, find $a_{2020}$. [b]p9.[/b] In $\vartriangle ABC$ with $AB = 10$ and $AC = 12$, points $D$ and $E$ lie on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = 4$ and $AE = 5$. If the area of quadrilateral $BCED$ is $40$, find the area of $\vartriangle ADE$. [b]p10.[/b] A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful? [b]p11.[/b] Let integers $A,B < 10, 000$ be the populations of Acton and Boxborough, respectively. When $A$ is divided by $B$, the remainder is $1$. When $B$ is divided by $A$, the remainder is $2020$. If the sum of the digits of $A$ is $17$, find the total combined population of Acton and Boxborough. [b]p12.[/b] Let $a_1$, $a_2$, $...$, $a_n$ be an increasing arithmetic sequence of positive integers. Given $a_n - a_1 = 20$ and $a^2_n - a^2_{n-1} = 63$, find the sum of the terms in the arithmetic sequence. [b]p13.[/b] Bob rolls a cubical, an octahedral and a dodecahedral die ($6$, $8$ and $12$ sides respectively) numbered with the integers from $1$ to $6$, $1$ to $8$ and $1$ to $12$ respectively. If the probability that the sum of the numbers on the cubical and octahedral dice equals the number on the dodecahedral die can be written as $\frac{m}{n}$ , where $m, n$ are relatively prime positive integers, compute $n - m$. [b]p14.[/b] Let $\vartriangle ABC$ be inscribed in a circle with center $O$ with $AB = 13$, $BC = 14$, $AC = 15$. Let the foot of the perpendicular from $A$ to BC be $D$ and let $AO$ intersect $BC$ at $E$. Given the length of $DE$ can be expressed as $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $m + n$. [b]p15.[/b] The set $S$ consists of the first $10$ positive integers. A collection of $10$ not necessarily distinct integers is chosen from $S$ at random. If a particular number is chosen more than once, all but one of its occurrences are removed. Call the set of remaining numbers $A$. Let $\frac{a}{b}$ be the expected value of the number of the elements in $A$, where $a, b$ are relatively prime positive integers. Find the reminder when $a + b$ is divided by $1000$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].