In a trapezoid $ABCD$, $AD \parallel BC$ and $\angle A = 60^o$. Let $E$ be a point on $AB$, and let $O_1$ and $O_2$ be circumcenters of $\vartriangle AED$ and $\vartriangle BEC$, respectively. Let $\frac{\overline{O_1O_2}}{\overline{DC}}$ be $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$? [asy]import three; size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); currentprojection=oblique; path3 p1=(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(0,2,2)--(0,0,2)--(2,0,2)--(2,2,2); path3 p2=(2,2,0)--(2,0,0)--(2,0,2); path3 p3=(0,0,2)--(0,2,1)--(2,2,1)--(2,0,2); path3 p4=(2,2,1)--(2,0,0); pen finedashed=linetype("4 4"); draw(p1^^p2^^p3^^p4); draw(shift((4,0,0))*p1); draw(shift((4,0,0))*p2); draw(shift((4,0,0))*p3); draw(shift((4,0,0))*p4); draw((4,0,2)--(5,2,2)--(6,0,2),finedashed); draw((5,2,2)--(5,2,0)--(6,0,0),finedashed); label("$W$",(3,0,2)); draw((2.7,.3,2)--(2.1,1.9,2),linewidth(.6pt)); draw((3.4,.3,2)--(5.9,1.9,2),linewidth(.6pt)); label("Figure 1",(1,-0.5,2)); label("Figure 2",(5,-0.5,2));[/asy]$ \textbf{(A)}\ \frac {1}{12}\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$

There are 8 members in a a bridge committee (committee for making bridges). Of these 8 members, 3 are chosen to be in special "approval" committee with 1 of 3 members being the "boss." In how many ways can this happen?

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

Consider an isosceles triangle $ABC$ with $AB=AC$. Point $D$ is on the smaller arc $AB$ of its circumcirle. Point $E$ lies on the continuation of segment $AD$ beyond point $D$ so that both $A$ and $E$ lie in the same half-plane relative to $BC$. The circumcirle of triangle $BDE$ intersects side $AB$ at point $F$. Prove that lines $EF$ and $BC$ are parallel. (Author: R. Zhenodarov)

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2\perp A_2B_1$, then $A_1B_2 = A_2B_1$.

Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$

A right cylinder is inscribed in a right circular cone with height $2$ and radius $2$ so that the cylinder’s bottom base sits on the cone’s base. What is the maximum possible surface area of the cylinder?

The area of the quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and $(-4,2,5)$ is the number $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

The winners of first AGMC in 2019 gifts the person in charge of the organiser, which is a polyhedron formed by $60$ congruent triangles. From the photo, we can see that this polyhedron formed by $60$ quadrilateral spaces. (Note: You can find the photo in 3.4 of [url]https://files.alicdn.com/tpsservice/18c5c7b31a7074edc58abb48175ae4c3.pdf?spm=a1zmmc.index.0.0.51c0719dNAbw3C&file=18c5c7b31a7074edc58abb48175ae4c3.pdf[/url]) A quadrilateral space is the plane figures that we fold the figures following the diagonal on a $n$ sides on a plane (i.e. form an appropriate dihedral angle in where the chosen diagonal is). "Two figure spaces are congruent" means they can coincide completely by isometric transform in $\mathbb{R}^3$. A polyhedron is the bounded space region, whose boundary is formed by the common edge of finite polygon. (a) We know that $2021=43\times 47$. Does there exist a polyhedron, whose surface can be formed by $43$ congruent $47$-gon? (b) Prove your answer in (a) with logical explanation.

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Let $ABCD$ a convex quadrilateral. Suppose that the circumference with center $B$ and radius $BC$ is tangent to $AD$ in $F$ and the circumference with center $A$ and radius $AD$ is tangent to $BC$ in $E$. Prove that $DE$ and $CF$ are perpendicular.

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[ 12x^2 + bxy + cy^2 + d = 0. \] Find the product $bc$.

There are $2n+1$ segments on the line. Any segment intersects at with at least $n$ others. Prove that there is a segment that intersects all the others.

(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively. (2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.

The shaded region formed by the two intersecting perpendicular rectangles, in square units, is [asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label("2",(0,.9),W); label("3",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label("10",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label("8",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1));[/asy] $ \text{(A)}\ 23\qquad\text{(B)}\ 38\qquad\text{(C)}\ 44\qquad\text{(D)}\ 46\qquad\text{(E)}\ \text{unable to be determined from the information given} $

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this. For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$. Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.