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

Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\Omega$. The line passing $I$ and perpendicular to $AI$ meets $AB, AC$ at $D, E$, respectively. $A$-excircle of $\triangle{ABC}$ meets $BC$ at $T$. $AT$ meets $\Omega$ at $P$. The line passing $P$ and parallel to $BC$ meets $\Omega$ at $Q$. The intersection of $QI$ and $AT$ is $K$. Prove that $Q,D,K,E$ are concyclic.

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

Let $ABCD$ be a square (vertices labelled in clockwise order). Let $Z$ be any point on diagonal $AC$ between $A$ and $C$ such that $AZ > ZC$. Points $X$ and $Y$ exist such that $AXY Z $ is a square (vertices labelled in clockwise order) and point $B$ lies inside $AXY Z$. Let $M$ be the point of intersection of lines $BX$ and $DZ$ (extended if necessary). Prove that $C$, $M$ and $Y$ are colinear

In plane there are two circles with radiuses $r_1$ and $r_2$, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points $A$ and $B$ and touches one of the circles in point $C$. Prove that $AC \cdot BC=r_1\cdot r_2$

On side $\overline{AB}$ of square $ABCD$, point $E$ is selected. Points $F$ and $G$ are located on sides $\overline{AB}$ and $\overline{AD}$, respectively, such that $\overline{FG} \perp \overline{CE}$. Let $P$ be the intersection point of segments $\overline{FG}$ and $\overline{CE}$. Given that $[EPF] = 1$, $[EPGA] = 8$, and $[CPFB] = 15$, compute $[PGDC]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.) [i]Proposed by Aaron Lin[/i]

Triangle $ABC$ satisfies $\angle ABC < \angle BCA < \angle CAB < 90^{\circ}$. $O$ is the circumcenter of triangle $ABC$, and $K$ is the reflection of $O$ in $BC$. $D,E$ is the foot of perpendicular line from $K$ to line $AB$, $AC$, respectively. Line $DE$ meets $BC$ at $P$, and a circle with diameter $AK$ meets the circumcircle of triangle $ABC$ at $Q(\neq A)$. If $PQ$ cuts the perpendicular bisector of $BC$ at $S$, then prove that $S$ lies on the circle with diameter $AK$.

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In this exercise we only consider convex quadrilaterals. (a) For such a quadrilateral $ABCD$, determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas. (b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$. (c) Find out which quadrilaterals $ABCD$ contains a point $P$, so that the triangles $PAB$, $PBC$, $PCD$ and $PDA$ have equal areas.

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.

Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.

In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP // AC, NQ // AB$, and $BP \cdot AC = CQ \cdot AB$. Find $\angle BAC$.

Let $T$ be any triangle such that its side-lengths $a, b$ and $c$ and its circumradius $R$ are positive integers. Show that: a) the inradius $r$ of $T$ is a positive integer; b) the perimeter $P$ of $T$ is a multiple of $4$; and c) all three of $a, b$ and $c$ are even.

Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $$ \min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $$

Let $ABCD$ be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides $ (AB+CD=AD+BC) $. Prove that the circles inscribed in triangles $ABC$ and $ACD$ are tangent to each other.

Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$. [b](a)[/b] Prove that any two of the following statements imply the third. [list] [b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$. [b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$. [b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list] [b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

in an acute triangle $ABC$,$D$ is a point on $BC$,let $Q$ be the intersection of $AD$ and the median of $ABC$from $C$,$P$ is a point on $AD$,distinct from $Q$.the circumcircle of $CPD$ intersects $CQ$ at $C$ and $K$.prove that the circumcircle of $AKP$ passes through a fixed point differ from $A$.

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.

Let $ABC$ be a triangle with $AC = 18$ and $D$ is the point on the segment $AC$ such that $AD = 5$. Draw perpendiculars from $D$ on $AB$ and $BC$ which have lengths $4$ and $5$ respectively. Find the area of the triangle $ABC$.

Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$. [i]Proposed by Lewis Chen[/i]