Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$. a) Prove that the lines $AC$ and $DE$ are parallel b) Prove that $AE=CD$

let $ H $ the orthocenter of the triangle $ ABC $ pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $ we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $ also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $ let $ d_1 \cap d_2 = Y $ prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line

A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$. [i](5 points)[/i]

We are given an arbitrary $\bigtriangleup ABC$. a) Can we dissect $\bigtriangleup ABC$ in $4$ pieces, from which we can make two triangle similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer $n \ge 2$ , we are able to dissect $\bigtriangleup ABC$ in $2n$ pieces, from which we can make two triangles similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer!

Let $ABC$ be an acute triangle and let $P, Q$ lie on the segment $BC$ such that $BP=PQ=CQ$. The feet of the perpendiculars from $P, Q$ to $AC, AB$ are $X, Y$. Show that the centroid of $ABC$ is equidistant from the lines $QX$ and $PY$.

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Let $ABCD$ be a parallelogram $.$ Let $O$ be a point in its interior such that $\angle AOB + \angle DOC = 180^{\circ} . $ Show that $,\angle ODC = \angle OBC . $

Let $\triangle MAB$ be a triangle with circumcenter $O$. $P$ and $Q$ lie on line $AB$ (both interior or exterior) such that $\angle PMA = \angle BMQ$. Let $D$ be a point on the perpendicular line through $M$ to $AB$. $E$ is the second intersection of the two circles $(DAB)$ and $(DPQ)$. The line $MO$ intersects $AB$ at $J$. Show that the circumcenter of $\triangle EMJ$ lies on line $AB$. [i](Proposed by Tan Rui Xuen)[/i]

Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $ m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is $ \textbf{(A)}\ \frac {1}{2m \plus{} 1} \qquad \textbf{(B)}\ m \qquad \textbf{(C)}\ 1 \minus{} m \qquad \textbf{(D)}\ \frac {1}{4m} \qquad \textbf{(E)}\ \frac {1}{8m^2}$

Let $ABC$ an acute-angle triangle. Let $R$ be a rectangle with vertices in the edges of $ABC$. Let $O$ be the center of $R$. a) Find the locus of all the points $O$. b) Decide if there is a point that is the center of three of these rectangles.

Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$, the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$. (a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle. (b) Find the area of quadrilateral $ABCD$.

Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$, and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at $F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals $AEFA', BDFB', CDEC'$ are inscribable. (1) Prove that $DEA'B'$ is inscribable. (2) Prove that $DA', EB', FC'$ are concurrent.

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

Let $A_1A_2A_3...A_n$ be a bicentric polygon with $n$ sides. Denote by $I_i$ the incenter of triangle $A_{i-1}A_iA_{i+1}, A_{i(i+1)}$ the intersection of $A_iA_{i+2}$ and $A_{i-1}A_{i+1},I_{i(i+1)}$ is the incenter of triangle $A_iA_{i(i+1)}A_{i+1}$ ($i = 1, n$). Prove that there exist $2n$ points $I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1}$ on the same circle. Nguyễn Văn Linh

Let $ABC$ be a triangle, and let $ABB_2A_3$, $BCC_3B_1$ and $CAA_1C_2$ be squares constructed outside the triangle. Denote with $S$ the area of the triangle $ABC$ and with s the area of the triangle formed by the intersection of the lines $A_1B_1$, $B_2C_2$ and $C_3A_3$. Prove that $s \le (4 - 2\sqrt3)S$.

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.