Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.

All diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? $\textbf{(A)} \ 49 \qquad \textbf{(B)} \ 65 \qquad \textbf{(C)} \ 70 \qquad \textbf{(D)} \ 96 \qquad \textbf{(E)} \ 128$

Let scalene triangle $ABC$ have altitudes $AD, BE, CF$ and circumcenter $O$. The circumcircles of $\triangle ABC$ and $\triangle ADO$ meet at $P \ne A$. The circumcircle of $\triangle ABC$ meets lines $PE$ at $X \ne P$ and $PF$ at $Y \ne P$. Prove that $XY \parallel BC$. [i]Proposed by Daniel Hu[/i]

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced. Prove that $QO$ is perpendicular to $BC$.

Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.

Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that \[ |p\cdot\overrightarrow{OA} \plus{} q\cdot\overrightarrow{OB} \plus{} r\cdot\overrightarrow{OC}|<\frac{1}{2007}\]

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

Consider the plane vectors $\overrightarrow{OA_1},\overrightarrow{OA_2},\dots ,\overrightarrow{OA_n}$ with $n\geq 3$. Suppose that the inequality $$\big|\overrightarrow{OA_1}+\overrightarrow{OA_2}+\dots +\overrightarrow{OA_n}\big|\geq \big|\pm\overrightarrow{OA_1}\pm\overrightarrow{OA_2}\pm\dots \pm\overrightarrow{OA_n}\big|$$ takes place for all choiches of the $\pm$ signs. Show that there exists a line $\ell$ through $O$ such that all points $A_1,A_2,\dots ,A_n$ are all on one side of $\ell$. [i]Cristi Săvescu[/i]

All the sides of the convex hexagon $ABCDEF$ are equal. In addition, $AD = BE = CF$. Prove that a circle can be inscribed into this hexagon. [i](Boyan Obukhov)[/i]

On the semicircle with diameter $AB$ and center $O$ point $D$ is marked. Points $E$ and $F$ are the midpoints of minor arcs $AD$ and $BD$ respectively. It turned out that the line connecting orthocenters of $ADF$ and $BDE$ passes through $O$ Find $\angle AOD$

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

[u]Round 5[/u] [b]p13.[/b] Express the number $3024_8$ in base $2$. [b]p14.[/b] $\vartriangle ABC$ has a perimeter of $10$ and has $AB = 3$ and $\angle C$ has a measure of $60^o$. What is the maximum area of the triangle? [b]p15.[/b] A weighted coin comes up as heads $30\%$ of the time and tails $70\%$ of the time. If I flip the coin $25$ times, howmany tails am I expected to flip? [u]Round 6[/u] [b]p16.[/b] A rectangular box with side lengths $7$, $11$, and $13$ is lined with reflective mirrors, and has edges aligned with the coordinate axes. A laser is shot from a corner of the box in the direction of the line $x = y = z$. Find the distance traveled by the laser before hitting a corner of the box. [b]p17.[/b] The largest solution to $x^2 + \frac{49}{x^2}= 2018$ can be represented in the form $\sqrt{a}+\sqrt{b}$. Compute $a +b$. [b]p18.[/b] What is the expected number of black cards between the two jokers of a $54$ card deck? [u]Round 7[/u] p19. Compute ${6 \choose 0} \cdot 2^0 + {6 \choose 1} \cdot 2^1+ {6 \choose 2} \cdot 2^2+ ...+ {6 \choose 6} \cdot 2^6$. [b]p20.[/b] Define a sequence by $a_1 =5$, $a_{n+1} = a_n + 4 * n -1$ for $n\ge 1$. What is the value of $a_{1000}$? [b]p21.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B = 15^o$ and $\angle C = 30^o$. Let $D$ be the point such that $\vartriangle ADC$ is an isosceles right triangle where $D$ is in the opposite side from $A$ respect to $BC$ and $\angle DAC = 90^o$. Find the $\angle ADB$. [u]Round 8[/u] [b]p22.[/b] Say the answer to problem $24$ is $z$. Compute $gcd (z,7z +24).$ [b]p23.[/b] Say the answer to problem $22$ is $x$. If $x$ is $1$, write down $1$ for this question. Otherwise, compute $$\sum^{\infty}_{k=1} \frac{1}{x^k}$$ [b]p24.[/b] Say the answer to problem $23$ is $y$. Compute $$\left \lfloor \frac{y^2 +1}{y} \right \rfloor$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a triangle $ABC$ with symmedians $BE,CF(E,F$ are on the sides $CA,AB,$ respectively$)$ intersecting at [i]Lemoine[/i] point $L.$ Prove that: $AB=AC$ in each case: a) $LB=LC.$ b) $BE=CF.$

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.

Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.

Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

The point $ (\minus{}3, 2)$ is rotated $ 90^\circ$ clockwise around the origin to point $ B$. Point $ B$ is then reflected over the line $ y \equal{} x$ to point $ C$. What are the coordinates of $ C$? $ \textbf{(A)}\ ( \minus{} 3, \minus{} 2)\qquad \textbf{(B)}\ ( \minus{} 2, \minus{} 3)\qquad \textbf{(C)}\ (2, \minus{} 3)\qquad \textbf{(D)}\ (2,3)\qquad \textbf{(E)}\ (3,2)$

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and \[\frac{|AC|}{|CB|}=\frac{3}{4}.\] The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.

Let $M$ be the midpoint of the cathetus $AC$ of a right-angled triangle $ABC$ $(\angle C=90^{\circ})$. The perpendicular from $M$ to the bisector of angle $ABC$ meets $AB$ at point $N$. Prove that the circumcircle of triangle $ANM$ touches the bisector of angle $ABC$. Proposed by:D.Shvetsov

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.