A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$

Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality: $$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that a) $\triangle OMN \sim \triangle OBA$; b) $OE \perp AB$. [i]Claudiu-Stefan Popa[/i]

Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that $$m(\widehat{ACB})=m(\widehat{PCT})$$

Euclid has a tool called [i]cyclos[/i] which allows him to do the following: [list] [*] Given three non-collinear marked points, draw the circle passing through them. [*] Given two marked points, draw the circle with them as endpoints of a diameter. [*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle. [/list] Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos. [i]Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu[/i]

Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

Two teams, $ A$ and $ B$, fight for a territory limited by a circumference. $ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved. Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas. Prove that, for every $ n$, team $ B$ has a winning strategy.

Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

Given a triangle $ABC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.

Which of these four numbers $ \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{\minus{}1}\cdot \sqrt{(.09)^{\minus{}1}}$, is (are) rational: $ \textbf{(A)}\ \text{none}\qquad \textbf{(B)}\ \text{all}\qquad \textbf{(C)}\ \text{the first and fourth}\qquad \textbf{(D)}\ \text{only the fourth}\qquad \textbf{(E)}\ \text{only the first}$

Five points $A$, $B$, $C$, $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$. Let $k$ be a circle tangent to $AD$, $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$, $B$ and $C$. Let $F \neq A$ be the intersection of $\tau$ and the tangent line to $k$ passing through $A$ different from $AD$. Prove that there exists a circle tangent to $BD$, $BF$, $CE$ and $\tau$.

In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.) [i]Proposed by Jonathan Liu[/i]

Let $ABC$ be a triangle inscribed in circle $(O)$ with diamter $KL$ passes through the midpoint $M$ of $AB$ such that $L, C$ lie on the different sides respect to $AB$. A circle passes through $M, K $cuts $LC$ at$ P, Q $(point $P$ lies between$ Q, C$). The line $KQ $cuts $(LMQ)$ at $R$. Prove that $ARBP$ is cyclic and$ AB$ is the symmedian of triangle $APR$. Please help :)

Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA \equal{} M$ and $ AQ \cap BC\equal{}N$, show that $ MN$, $ r$ and $ AC$ concur.

Let $K$ be the intersection of diagonals of cyclic quadrilateral $ABCD$, where $|AB|=|BC|$, $|BK|=b$, and $|DK|=d$. What is $|AB|$? $ \textbf{(A)}\ \sqrt{d^2 + bd} \qquad\textbf{(B)}\ \sqrt{b^2+bd} \qquad\textbf{(C)}\ \sqrt{2bd} \qquad\textbf{(D)}\ \sqrt{2(b^2+d^2-bd)} \qquad\textbf{(E)}\ \sqrt{bd} $

Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.

The length $l$ of a tangent, drawn from a point $A$ to a circle, is $\frac43$ of the radius $r$. The (shortest) distance from $A$ to the circle is: $ \textbf{(A)}\ \frac{1}{2}r \qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \frac{1}{2}l\qquad\textbf{(D)}\ \frac23l \qquad\textbf{(E)}\ \text{a value between r and l.} $