A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

The plane is covered with circles in such a way that the center of each circle does not belong to any other circle. Prove that each point of the plane belongs to at most five circles.

Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$

Let $\vartriangle ABC$ be an acute triangle with altitudes $AA_1, BB_1, CC_1$ and orthocenter $H$. Let $K, L$ be the midpoints of $BC_1, CB_1$. Let $\ell_A$ be the external angle bisector of $\angle BAC$. Let $\ell_B, \ell_C$ be the lines through $B, C$ perpendicular to $\ell_A$. Let $\ell_H$ be the line through $H$ parallel to $\ell_A$. Prove that the centers of the circumcircles of $\vartriangle A_1B_1C_1, \vartriangle AKL$ and the rectangle formed by $\ell_A, \ell_B, \ell_C, \ell_H$ lie on the same line.

Let $ABCD$ be a convex quadrilateral with angles $\sphericalangle A, \sphericalangle C\geq90^{\circ}$. On sides $AB,BC,CD$ and $DA$, consider the points $K,L,M$ and $N$ respectively. Prove that the perimeter of $KLMN$ is greater than or equal to $2\cdot AC$.

$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$. a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value. b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$. c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.

Circles $\alpha$ and  $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to  $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$. (D. Mukhin)

On the plane are chosen $n \ge 3$ points, not all on the same line. Drawing all lines passing through two of these points one obtains k different lines. Prove that $k \ge n$.

Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

Prove that if the cross-section of a cube cut by a plane is a pentagon, as shown in the figure below, then there are two adjacent sides of the pentagon such that the sum of the lengths of those two sides is greater than the sum of the lengths of the other three sides. (For ease of grading, please use the names of the points from the figure below in your solution.) [asy] import three; defaultpen(linewidth(0.8)); currentprojection=orthographic(1,3/5,1/2); draw(unitcube, white, thick(), nolight); draw(O--(1,0,0)^^O--(0,1,0)^^O--(0,0,1), linetype("4 4")+linewidth(0.7)); triple A=(1/3, 1, 1), B=(2/3, 1, 0), C=(1, 1/2, 0), D=(1, 0, 1/2), E=(2/3, 0, 1); draw(E--A--B^^C--D); draw(B--C^^D--E, linetype("4 4")+linewidth(0.7)); label("$A$", A, dir(85)); label("$B$", B, SE); label("$C$", C, S); label("$D$", D, W); label("$E$", E, NW);[/asy]

Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases [b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$. [b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$. In other words, find the Steiner trees of the set $M$ in the above two cases.

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_j$ have a common point for all $i, j \in \{1,..., n\}$, $i \ne j$. [i]Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices [/i]

Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have \[ \left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K. \] [i]2015 CCA Math Bonanza Lightning Round #3.4[/i]

Points $P$ and $Q$ inside a triangle $ABC$ with sides $a,b,c$ and the corresponding angle $\alpha,\beta,\gamma$ satisfy $\angle BPC=\angle CPA=\angle APB=120^\circ$ and $\angle BQC=60^\circ+\alpha$, $\angle CQA=60^\circ+\beta$, $\angle AQB=60^\circ+\gamma$. Prove the equality $$(AP+BP+CP)^3\cdot AQ\cdot BQ\cdot CQ=(abc)^2.$$

A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even. [img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]

Let $A, B, C, D$ be points chosen on a circle, in that order. Line $BD$ is reflected over lines $AB$ and $DA$ to obtain lines $\ell_1$ and $\ell_2$ respectively. If lines $\ell_1$, $\ell_2$, and $AC$ meet at a common point and if $AB = 4$, $BC = 3$, $CD = 2$, compute the length $DA$.

In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.

Through points $ A $ and $ B $ two oblique lines $ m $ and $ n $ are drawn perpendicular to the line $ AB $. On line $ m $ the point $ C $ (different from $ A $) is taken, and on line $ n $ the point $ D $ (different from $ B $) is taken. Given the lengths of segments $ AB = d $ and $ CD = l $ and the angle $ \varphi $ formed by the oblique lines $ m $ and $ n $, calculate the radius of the surface of the sphere passing through the points $ A $, $ B $, $ C $, $ D $.

A point $O$ is chosen inside the square $ABCD$. The square $A'B'C'D'$ is the image of the square $ABCD$ under the homothety with center at point $O$ and coefficient $k> 1$ (points $A', B', C', D' $ are images of points $A, B, C, D$ respectively). Prove that the sum of the areas of the quadrilaterals $A'ABB'$ and $C'CDD'$ is equal to the sum of the areas quadrilaterals $B'BCC'$ and $D'DAA'$.

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Let $D$ be a point on the side $AC$ of a triangle $ABC$. Suppose that the incircle of triangle $BCD$ intersects $BD$ and $CD$ at $X$, $Y$, respectively. Show that $XY$ passes through a fixed point when $D$ is moving on the side $AC$.