Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$ Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.

The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.

Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.

Given three equidistant parallel lines. Express by points of the corresponding lines the values of the resistance, voltage and current in a conductor so as to obtain the voltage $V = I \cdot R$ by connecting with a ruler the points denoting the resistance $R$ and the current $I$. (Each point of each scale denotes only one number). [hide=similar wording]Three parallel straight lines are given at equal distances from each other. How to depict by points of the corresponding straight lines the values of resistance, voltage and the current in the conductor, so that, applying a ruler to to points depicting the values of resistance R and values of current I, obtain on the voltage scale a point depicting the value of voltage V = I R (point each scale represents one and only one number).[/hide]

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.

The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.

In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.

Given $\vartriangle ABC$, points $C_1, A_1, B_1$ on sides $AB, BC, CA$, respectively, such that $\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n}$ and points $C_2, A_2, B_2$ on sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, such that $\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n$. Prove that $A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA$.

Let $H$ be the orthocenter of acute-angled $\vartriangle ABC$, and $X, Y$ points on the ray $AB, AC$. ($B$ lies between $X, A$, and $C$ lies between $Y, A$.) Lines $HX, HY$ intersect $BC$ at $D, E$ respectively. Let the line through $D$ parallel to $AC$ intersect $XY$ at $Z$. Prove that $\angle XHY = 90^o$ if and only if $ZE \parallel AB$.

In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng