On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

On the inside of the triangle $ABC$ a point $P$ is chosen with $\angle BAP = \angle CAP$. If $\left | AB \right |\cdot \left | CP \right |= \left | AC \right |\cdot \left | BP \right |= \left | BC \right |\cdot \left | AP \right |$ , find all possible values of the angle $\angle ABP$.

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$? [asy] import cse5;pathpen=black;pointpen=black; size(2inch); pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162); D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5)); D(A--C--E--B--D--cycle); pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A); D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle); [/asy] $\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10} $

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy] unitsize(.85cm); draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1)); draw((0,2)--(0,1)--(1,1)--(1,2),dashed); label("\footnotesize $a$",(0.5,0),S); label("\footnotesize $a$",(0,0.5),W); label("\footnotesize $a$",(1,0.5),E); label("\footnotesize $a$",(0,1.5),E); label("\footnotesize $a$",(1,1.5),W); label("\footnotesize $a$",(-0.5,2),N); label("\footnotesize $a$",(0.5,2),N); label("\footnotesize $a$",(1.5,2),N); [/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length $a$ of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of $\lambda$ such that every such configuration of tiles has a total area less than $\lambda$.

Let $\omega$ and $\Omega$ be circles of radius $1$ and $R>1$ respectively that are internally tangent at a point $P$. Two tangent lines to $\omega$ are drawn such that they meet $\Omega$ at only three points $A$, $B$, and $C$, none of which are equal to $P$. If triangle $ABC$ has side lengths in a ratio of $3:4:5$, find the sum of all possible values of $R$. [i]Proposed by Connor Gordon[/i]

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.

Two non-congruent triangles are called [i]analogous [/i] if they can be denoted as $ABC$ and $A'B'C'$ such that $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ . Do there exist three mutually [i]analogous[/i] triangles?

Circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects the circle $k_1$ at points $A$ and $C$, the circle $K_2$ at points $B$ and $D$ so that the points $A,B,C$ and $D$ lie on the line $\ell$ are in that order. Let $X$ a point on the line $MN$ such that the point $M$ is located between the points $X$ and $N$. Let $P$ be the intersection of lines $AX$ and $BM$, and $Q$ be the intersection of lines $DX$ and $CM$. If $K$ is the midpoint of segment $AD$ and $L$ is the midpoint of segment $BC$, prove that the lines $XK$ and $ML$ intersect on the line $PQ$.

$H$ is the orthocenter of a non-isosceles acute triangle $\triangle ABC$. $M$ is the midpoint of $BC$ and $BB_1, CC_1$ are two altitudes of $\triangle ABC$. $N$ is the midpoint of $B_1C_1$. Prove that $AH$ is tangent to the circumcircle of $\triangle MNH$.

The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?

Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?

[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$. [b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests. "I wonder how many knights are among you?" he asked. " Ask everyone a question and find out yourself" advised him one of the guests. "Okay. Tell me one: Who are your neighbors?" asked the traveler. This question was answered the same way by all the guests. "This information is not enough!" said the traveler. "But today is my birthday, do not forget it!" said one of the guests. "Yes, today is his birthday!" said his neighbor. Now the traveler was able to find out how many knights were at the table. Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]? [b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters? [b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed? [b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.

Let $\Gamma$ denote the circumcircle and $O$ the circumcentre of the acute-angled triangle $ABC$, and let $M$ be the midpoint of the segment $BC$. Let $T$ be the second intersection point of $\Gamma$ and the line $AM$, and $D$ the second intersection point of $\Gamma$ and the altitude from $A$. Let further $X$ be the intersection point of the lines $DT$ and $BC$. Let $P$ be the circumcentre of the triangle $XDM$. Prove that the circumcircle of the triangle $OPD$ passes through the midpoint of $XD$.

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.

Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.

$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$. $O$ meets the segment $AB$ at $D$. $CO$ meets the circle $(O)$ again at $P$, a line, which passes $P$ and parallel to $AO$, meets $AC$ at $E$, and $EB$ meets the circle $(O)$ again at $L$. A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$. Prove that two lines $EK$ and $CL$ are parallel.

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$. [i]Proposed by Mehran Talaei[/i]

Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.

Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.