While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.

Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?

Points $E, F, G$ lie, and on the sides $BC, CA, AB$, respectively of a triangle $ABC$, with $2AG=GB, 2BE=EC$ and $2CF=FA$. Points $P$ and $Q$ lie on segments $EG$ and $FG$, respectively such that $2EP = PG$ and $2GQ=QF$. Prove that the quadrilateral $AGPQ$ is a parallelogram.

Let $ABC$ be isosceles triangle ($AB=BC$) and $K$ and $M$ be the midpoints of $AB$ and $AC,$ respectively.Let the circumcircle of $\triangle BKC$ meets the line $BM$ at $N$ other than $B.$ Let the line passing through $N$ and parallel to $AC$ intersects the circumcircle of $\triangle ABC$ at $A_1$ and $C_1.$ Prove that $\triangle A_1BC_1$ is equilateral.

Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.

Quadrilateral $ABCD$ is cyclic and has positive integer side lengths. Suppose $AC \cdot BD = 53$ and $CD < DA$. The value of $\frac{AB /BC}{AD /DC}$ can be expressed as a common fraction $\frac{p}{q}$ , where $p$ and $q$ are relatively prime. Compute $p + q$.

A circle has radius $52$ and center $O$. Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$. Point $Q$ is constructed such that $QA = QP = 15$, and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$. Find $QB$. [i]Proposed by Justin Hsieh[/i]

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the region bounded by the lines $y = x/2$, $y = -x/2$, and $x = 6$. Pick a random point $P = (x, y)$ in $S$ and translate it $3$ units right to $P' = (x + 3, y)$. What is the probability that $P'$ is in $S$? [b]p26.[/b] A triangle with side lengths $17$, $25$, and $28$ has a circle centered at each of its three vertices such that the three circles are mutually externally tangent to each other. What is the combined area of the circles? [b]p27.[/b] Find all ordered pairs $(x, y)$ of integers such that $x^2 - 2x + y^2 - 6y = -9$. [u]Round 10[/u] [b]p28.[/b] In how many ways can the letters in the word $SCHAFKOPF$ be arranged if the two $F$’s cannot be next to each other and the $A$ and the $O$ must be next to each other? [b]p29.[/b] Let a sequence $a_0, a_1, a_2, ...$ be defined by $a_0 = 20$, $a_1 = 11$, $a_2 = 0$, and for all integers $n \ge 3$, $$a_n + a_{n-1 }= a_{n-2} + a_{n-3}.$$ Find the sum $a_0 + a_1 + a_2 + · · · + a_{2010} + a_{2011}$. [b]p30.[/b] Find the sum of all positive integers b such that the base $b$ number $190_b$ is a perfect square. [u]Round 11[/u] [b]p31.[/b] Find all real values of x such that $\sqrt[3]{4x -1} + \sqrt[3]{4x + 1 }= \sqrt[3]{8x}$. [b]p32.[/b] Right triangle $ABC$ has a right angle at B. The angle bisector of $\angle ABC$ is drawn and extended to a point E such that $\angle ECA = \angle ACB$. Let $F$ be the foot of the perpendicular from $E$ to ray $\overrightarrow{BC}$. Given that $AB = 4$, $BC = 2$, and $EF = 8$, find the area of triangle $ACE$. [b]p33.[/b] You are the soul in the southwest corner of a four by four grid of distinct souls in the Fields of Asphodel. You move one square east and at the same time all the other souls move one square north, south, east, or west so that each square is now reoccupied and no two souls switched places directly. How many end results are possible from this move? [u]Round 12[/u] [b]p34.[/b] A [i]Pythagorean [/i] triple is an ordered triple of positive integers $(a, b, c)$ with $a < b < c $and $a^2 + b^2 = c^2$ . A [i]primitive [/i] Pythagorean triple is a Pythagorean triple where all three numbers are relatively prime to each other. Find the number of primitive Pythagorean triples in which all three members are less than $100,000$. If $P$ is the true answer and $A$ is your team’s answer to this problem, your score will be $max \left\{15 -\frac{|A -P|}{500} , 0 \right\}$ , rounded to the nearest integer. [b]p35.[/b] According to the Enable2k North American word list, how many words in the English language contain the letters $L, M, T$ in order but not necessarily together? If $A$ is your team’s answer to this problem and $W$ is the true answer, the score you will receive is $max \left\{15 -100\left| \frac{A}{W}-1\right| , 0 \right\}$, rounded to the nearest integer. [b]p36.[/b] Write down $5$ positive integers less than or equal to $42$. For each of the numbers written, if no other teams put down that number, your team gets $3$ points. Otherwise, you get $0$ points. Any number written that does not satisfy the given requirement automatically gets $0$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url]. Rest Rounds soon. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?

Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.

In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.

Let $ABCDEF$ be an equilateral heaxagon such that $\triangle ACE \cong \triangle DFB$. Given that $AC = 7$, $CE=8$, and $EA=9$, what is the side length of this hexagon? [i]Proposed by Thomas Lam[/i]

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

It is given a convex quadrilateral $ ABCD$ in which $ \angle B\plus{}\angle C < 180^0$. Lines $ AB$ and $ CD$ intersect in point E. Prove that $ CD*CE\equal{}AC^2\plus{}AB*AE \leftrightarrow \angle B\equal{} \angle D$

In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .

Circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$. Segments $AC$ and $BD$ are chords of $\omega_1$ and $\omega_2$ respectively, such that segment $AB$ and ray $CD$ meet at $P$. Ray $BD$ and segment $AC$ meet at $X$. Point $Y$ lies on $\omega_1$ such that $P Y \parallel BD$. Point $Z$ lies on $\omega_2$ such that $P Z \parallel AC$. Prove that points $Q,~ X,~ Y,~ Z$ are collinear.

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$.

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes. (N. Belukhov)

In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.

In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$ $ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

Let $ABCD$ be a square of side 5, $E$ a point on $BC$ such that $BE=3, EC= 2$. Let $P$ be a variable point on the diagonal $BD.$ Determine the length of $PB$ if $PE+PC$ is smallest.