The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters? [asy]/* AMC8 2003 #21 Problem */ size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype("8 4")); draw((11,1)--(12,1)--(12,0)); label("$A$", (0,0), SW); label("$D$", (31,0), SE); label("$B$", (6,8), NW); label("$C$", (16,8), NE); label("10", (3,5), W); label("8", (11,4), E); label("17", (22.5,5), E);[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $X$ and $K$ points over $AC$ and $AB$, respectively, such that $KX = CX$. Bisector of $\angle AKX$ intersects line $BC$ at $Z$. Show that $XZ$ passes through the midpoint of $BK$.

Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$. Prove that the lines $AT$ and $XY$ are parallel. [i]Navneel Singhal, India[/i]

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.

Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? [asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("$7$",(1.25,0.2)); label("$7$",(2.2,0.45)); label("$3$",(0.45,0.35));[/asy] $ \textbf{(A) }15\qquad\textbf{(B) }17\qquad\textbf{(C) }\frac{35}{2}\qquad\textbf{(D) }18\qquad\textbf{(E) }\frac{55}{3} $

a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively. Prove that: $M,O,N,K$ are on the same circle. b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$

In the star shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$ , where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and that $b$ is square-free. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/d16a78317b0298d6894c6bd62fbcd1a5894306.png[/img]

In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.

Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.

Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides.

$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.

An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: $\text{(A)}\ 16n\qquad \text{(B)}\ 18n\qquad \text{(C)}\ 21n\qquad \text{(D)}\ 24n\qquad \text{(E)}\ 27n$

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

Inside triangle \( ABC \), a point \( D \) is chosen such that \( \angle ADB = \angle ADC \). The rays \( BD \) and \( CD \) intersect the circumcircle of triangle \( ABC \) at points \( E \) and \( F \), respectively. On segment \( EF \), points \( K \) and \( L \) are chosen such that \linebreak \( \angle AKD = 180^\circ - \angle ACB \) and \( \angle ALD = 180^\circ - \angle ABC \), with segments \( EL \) and \( FK \) \linebreak not intersecting line \( AD \). Prove that \( AK = AL \). [i]Proposed by Matthew Kurskyi[/i]

The area $T$ and an angle $\gamma$ of a triangle are given. Determine the lengths of the sides $a$ and $b$ so that the side $c$, opposite the angle $\gamma$, is as short as possible.

Let $ABC$ be a triangle . Let $A'$ be the center of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$ , respectively ; the points $B'$ and $C'$ are defined similarly . Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.

Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

Find the number of points $P \in Z^2$ that satisfy the following two conditions: 1) If $Q$ is a point on the circle of radius $\sqrt{2020}$ centered at the origin such that the line $PQ$ is tangent to the circle at $Q$, then $PQ$ has integral length. 2) The x-coordinate of $P$ is $38$.