Let $ABCDEF$ be a regular hexagon with side length $2$. Calculate the area of $ABDE$. [i]2015 CCA Math Bonanza Lightning Round #1.2[/i]

Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$. Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$. Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$). Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$). Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$. Find the length of $AP$.

Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure. $ \begin{tabular}{|c|c|c|c|c|c|} \hline & & & & & \\ \hline & & & & & \\ \hline & & \multicolumn{1}{c}{} & & & \\ \cline{1 \minus{} 2}\cline{5 \minus{} 6} & & \multicolumn{1}{c}{} & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.

Given is an acute triangle $ABC$ with $BC < CA < AB$. Points $K$ and $L$ lie on segments $AC$ and $AB$ and satisfy $AK = AL = BC$. Perpendicular bisectors of segments $CK$ and $BL$ intersect line $BC$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at $M$. Prove that $CK + KM = BL + LM$.

A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.

Given is natural number $n$. Sasha claims that for any $n$ rays in space, no two of which have a common point, he will be able to mark on these rays $k$ points lying on one sphere. What is the largest $k$ for which his statement is true?

Show that if the sides $a, b, c$ of a triangle satisfy the equation \[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\] then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.

Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.

Let $O$ be an interior point of an acute triangle $ABC$. The circles with centers the midpoints of its sides and passing through $O$ mutually intersect the second time at the points $K$, $L$ and $M$ different from $O$. Prove that $O$ is the incenter of the triangle $KLM$ if and only if $O$ is the circumcenter of the triangle $ABC$.

What is the area of the largest circle contained in an equilateral triangle of area $8\sqrt3$?

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that: \[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\] where R is the radius of circumcircle of $\triangle ABC$.

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively. If $AC,BD,EG,FH$ concur at a point $O,$ prove that $ABCD$ is a parallelogram.

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.

Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$. Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $P$ and $Q$. The tangent line closer to $Q$ touches $\omega_1$ and $\omega_2$ at $M$ and $N$ respectively. If $P Q = 3$, $QN = 2$, and $MN = P N$, what is $QM^2$?

A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? $ \textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255 $

Let a triangle $ABC$ be given. On a ruler three segment congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.

In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$? [i]Proposed by James Lin[/i]

Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that: a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$. b) $MN\le BD$. When the equality holds?