Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

Let $O{}$ be the circumcenter of an acute triangle $ABC$. Let $M{}$ be the midpoint of $AC$. The straight line $BO$ intersects the altitudes $AA_1{}$ and $CC_1{}$ at the points $H_a$ and $H_c$ respectively. The circumcircles of the triangles $BH_aA$ and $BH_cC$ have a second point of intersection $K{}$. Prove that $K{}$ lies on the straight line $BM$. [i]Mikhail Evdokimov[/i]

A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle. [i]J. Tabov[/i]

We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove thatthe points $P, Q, R, S$ are concyclic.

In a right triangle $ ABC $, the altitude $ CD $ is drawn from the vertex of the right angle $ C $ and a circle is inscribed in each of the triangles $ ABC $, $ ACD $ and $ BCD $. Prove that the sum of the radii of these circles equals the height $ CD $.

Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined: $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$ $O_n$ is the center of $\odot (PA_nB_n)$ $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$ $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$ If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ [hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

The triangle $ABC$ is given. Let $A'$ be the midpoint of the side $BC$, $B_c{}$ be the projection of $B{}$ onto the bisector of the angle $ACB{}$ and $C_b$ be the projection of the point $C{}$ onto the bisector of the angle $ABC$. Let $A_0$ be the center of the circle passing through $A', B_c, C_b$. The points $B_0$ and $C_0$ are defined similarly. Prove that the incenter of the triangle $ABC$ coincides with the orthocenter of the triangle $A_0B_0C_0$.

Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.

Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.

Let $ABC$ be a triangle with $\angle{A}=90^{\circ}$ and $AB<AC$. Let $AD$ be the altitude from $A$ on to $BC$, Let $P,Q$ and $I$ denote respectively the incentres of triangle $ABD,ACD$ and $ABC$. Prove that $AI$ is perpendicular to $PQ$ and $AI=PQ$.

A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$? [i]Proposed by Eugene Chen[/i]

Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

$ABCD$ is a cyclic quadrilateral. Lines $AB,CD$ intersect at $E$, lines $AD,BC$ intersect at $F$, and $EM$ and $FN$ are tangents to the circumcircle of $ABCD$. Two circles are constructed with $E,F$ their centers and $EM, FN$ their radii, respectively. $K$ is one of their intersections. Prove that $EK$ is perpendicular to $FK$.

A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. [img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]

Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.

Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34.$ Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$. [asy] size(200); defaultpen(linewidth(0.8)+fontsize(10.6)); pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", (H+E)/2,fontsize(15)); label("$x$", (E+F)/2,fontsize(15)); label("$y$", (G+F)/2,fontsize(15)); label("$z$", (H+G)/2,fontsize(15)); label("$w:x:y:z=269:275:405:411$",(sqrt(850)/2,-4.5),fontsize(11)); [/asy]

Let $S$ be the set of all tetrahedra which satisfy: (1) the base has area $1$, (2) the total face area is $4$, and (3) the angles between the base and the other three faces are all equal. Find the element of $S$ which has the largest volume.

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

Let a regular $ (2n \plus{}1)\minus{}$gon be inscribed in a circle of radius $ r.$ We consider all the triangles whose vertices are from those of the regular $ (2n \plus{} 1)\minus{}$gon. [b](a)[/b] How many triangles among them contain the center of the circle in their interior? [b](b)[/b] Find the sum of the areas of all those triangles that contain the center of the circle in their interior.