Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions: [b]I.[/b] $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$. [b]II.[/b] Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$. (By $(AB)$ we denote vector $AB$)

Let $ABCD$ be a parallelogram. A circle with diameter $AC$ intersects line $BD$ at points $P$ and $Q$. The perpendicular on $AC$ passing through point $C$, intersects lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P, Q, X$ and $Y$ lie on the same circle.

Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0$. Prove that $ MN$ is parallel to $ AB$.

The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$. Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.

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} $

The points $E$ and $F$ divide the diagonal $BD$ of the convex quadrilateral $ABCD$ into three equal parts, i.e. $| BE | = | EF | = | F D |$. Line $AE$ interects side $BC$ at $X$ and line $AF$ intersects $DC$ at $Y$. Prove that: a) if $ABCD$ is parallelogram then $X ,Y$ are the midpoints of $BC, DC$, respectively, b) if the points $X , Y$ are the midpoints of $BC, DC$, respectively , then $ABCD$ is parallelogram

Let $ABCD$ be a rhombus with $\angle DAB = 60^o$. Let $K, L$ be points on its sides $AD$ and $DC$ and $M$ a point on the diagonal $AC$ such that $KDLM$ is a parallelogram. Prove that triangle $BKL$ is equilateral.

On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram

1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions: [b]I.[/b] The distance between any two points $\in S$ is not greater than 1. [b]II.[/b] For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$. [b]III. [/b] For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$. Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Let $l,m$ be two parallel lines in the plane. Let $P$ be a fixed point between them. Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$. (By angle $EPF$ we mean the directed angle) Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.

Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is congruent to $\triangle H_AH_BH_C$.

In the parallelogram $ABCD$, the length of side $AB$ is half the length of side $BC$. The bisector of angle $\angle ABC$ intersects side $AD$ at point $K$ and diagonal $AC$ at point $L$. The bisector of angle $\angle ADC$ intersects the extension of side $AB$ at point $M$, with $B$ between $A$ and $M$. The line $ML$ intersects side $AD$ at point $F$. Calculate the ratio $\frac{AF}{AD}$.

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches. ${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

Starting from a given cyclic quadrilateral $\mathcal{Q}_0$, a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$, $B$, $C$ and $D$.) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

Points $D$ and $E$ are taken on sides $BC$ and $CA$ of a triangle $ BD\equal{}AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of $\angle ACB$ intersects $AD$ and $BE$ at $Q$ and $R$ respectively. Prove that $ \frac{PQ}{PR}\equal{}\frac{AD}{BE}$.

Let $l,m$ be two parallel lines in the plane. Let $P$ be a fixed point between them. Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$. (By angle $EPF$ we mean the directed angle) Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.

Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.

Let $ H$ be the orthocenter, $ O$ the circumcenter, and $ R$ the circumradius of an acute-angled triangle $ ABC$. Consider the circles $ k_a,k_b,k_c,k_h,k$, all with radius $ R$, centered at $ A,B,C,H,M,$ respectively. Circles $ k_a$ and $ k_b$ meet at $ M$ and $ F$; $ k_a$ and $ k_c$ meet at $ M$ and $ E$; and $ k_b$ and $ k_c$ meet at $ M$ and $ D$. $ (a)$ Prove that the points $ D,E,F$ lie on the circle $ k_h$. $ (b)$ Prove that the set of the points inside $ k_h$ that are inside exactly one of the circles $ k_a,k_b,k_c$ has the area twice the area of $ \triangle ABC$.

Given a parallelogram $ABCD$. Let $E$ be a point on the side $BC$ and $F$ be a point on the side $CD$ such that the triangles $ABE$ and $BCF$ have the same area. The diaogonal $BD$ intersects $AE$ at $M$ and intersects $AF$ at $N$. Prove that: [b]I. [/b] There exists a triangle, three sides of which are equal to $BM, MN, ND$. [b]II.[/b] When $E, F$ vary such that the length of $MN$ decreases, the radius of the circumcircle of the above mentioned triangle also decreases.