Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property: For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices. [hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]

Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following: $$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$

Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.

An $n$-dimensional cube is given. Consider all the segments connecting any two different vertices of the cube. How many distinct intersection points do these segments have (excluding the vertices)?

Let \(\Omega\) be a circle with a chord \(AB\) which is not a diameter. \(\Gamma_{1}\) be a circle on one side of \(AB\) such that it is tangent to \(AB\) at \(C\) and internally tangent to \(\Omega\) at \(D\). Likewise, let \(\Gamma_{2}\) be a circle on the other side of \(AB\) such that it is tangent to \(AB\) at \(E\) and internally tangent to \(\Omega\) at \(F\). Suppose the line \(DC\) intersects \(\Omega\) at \(X \neq D\) and the line \(FE\) intersects \(\Omega\) at \(Y \neq F\). Prove that \(XY\) is a diameter of \(\Omega\) .

Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle. [i]Costel Anghel[/i]

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

Let $X$ be the intersection of the diagonals $AC$ and $BD$ of convex quadrilateral $ABCD$. Let $P$ be the intersection of lines $AB$ and $CD$, and let $Q$ be the intersection of lines $PX$ and $AD$. Suppose that $\angle ABX = \angle XCD = 90^o$. Prove that $QP$ is the angle bisector of $\angle BQC$.

Two vertices of the rectangle are located on side $BC$ of triangle $ABC$, and the other two are on sides $AB$ and $AC$. It is known that the midpoint of the altitude of this triangle, drawn on the side $BC$, lies on one of the diagonals of the rectangle, and the side of the rectangle located on $BC$ is three times less than $BC$. In what ratio does the altitude of the triangle divide the side $BC$ ?

Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$. $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$. The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$. Prove that $A', D$ and $Z$ are collinear. [i]Proposed by Nikola Velov[/i]

Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$. Point $P$ lies in the exterior of the circle and does not lie on the line $AB$. Using only an unmarked straightedge, construct a line through $P$, perpendicular to the line $AB$. Give complete instructions for the construction and prove that it works.

[i]The Marathon.[/i] Let $\omega$ denote the incircle of triangle $ABC$. The segments $BC$, $CA$, and $AB$ are tangent to $\omega$ at $D$, $E$ and $F$, respectively. Point $P$ lies on $EF$ such that segment $PD$ is perpendicular to $BC$. The line $AP$ intersects $BC$ at $Q$. The circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$, respectively, and are tangent to $AQ$ at $Q$; the former meets $AB$ again at $X$, and the latter meets $AC$ again at $Y$. The line $XY$ intersects $BC$ at $Z$. Given that $AB=15$, $BC=14$, and $CA=13$, find $\lfloor XZ\cdot YZ\rfloor$.

Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?

Given an acute-angled triangle $ABC$, in which $P$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $AC$, respectively. A point $H$ is taken inside the triangle and perpendiculars $HK$, $HS$, $HQ$ are lowered from it to the sides $AB$, $BC$, $AC$, respectively ($K \in AB$, $S \in BC$, $Q \in AC$). It turned out that $MK = MQ$, $NS = NK$, $PS=PQ$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $

In $\triangle ABC$, $D,E,F$ are respectively the midpoints of $AB, BC, and CA$. Futhermore $AB=10$, $CD=9$, $CD\perp AE$. Find $BF$.

Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo: Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let $F_A$ be the (associated with $\Omega$) Feuerbach point of the triangle formed by the tangents to $\Omega$ at $B,C,D$, that is, the point of tangency of $\Omega$ and the nine-point circle of that triangle. Define $F_B, F_C, F_D$ similarly. Let $A'$ be the intersection of the tangents to $\Omega$ at $A$ and $F_A$. Define $B', C', D'$ similarly. Prove that quadrilaterals $ABCD$ and $A'B'C'D'$ are similar

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.