Two points $P$ and $Q$ lying on side $BC$ of triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $. Author:Mehdi E'tesami Fard , Iran

Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.

Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.

Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.

A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.

Circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) with centers at \(C_1\) and \(C_2\) respectively, intersect at two points \(A\) and \(B\). Points \(P\) and \(Q\) are varying points on \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively, such that \(P\), \(Q\) and \(B\) are collinear and \(B\) is always between \(P\) and \(Q\). Let lines \(PC_1\) and \(QC_2\) intersect at \(R\), let \(I\) be the incenter of \(\Delta PQR\), and let \(S\) be the circumcenter of \(\Delta PIQ\). Show that as \(P\) and \(Q\) vary, \(S\) traces the arc of a circle whose center is concyclic with \(A\), \(C_1\) and \(C_2\).

Let $D$ be a point inside a triangle $ABC$ such that $|AC| -|AD| \geq 1$ and $|BC|- |BD| \geq 1.$ Prove that for any point $E$ on the segment $AB$, we have $|EC| -|ED| \geq 1.$

Points $A$, $B$, and $C$ lie on a circle of radius $5$ such that $AB=6$ and $AC=8$. Find the smaller of the two possible values of $BC$.

Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.

The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.

Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$

[b]p7.[/b] Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles. [img]https://cdn.artofproblemsolving.com/attachments/9/7/7ff192baf58f40df0e4cfae4009836eab57094.png[/img] [b]p8.[/b] Find the greatest divisor of $40!$ that has exactly three divisors. [b]p9.[/b] Suppose we have positive integers $a, b, c$ such that $a = 30$, lcm $(a, b) = 210$, lcm $(b, c) = 126$. What is the minimum value of lcm $(a, c)$? PS. You should use hide for answers.

Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$). Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy [b](i)[/b] $M_0 = M_k$, and [b](ii)[/b] if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.

A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.

Point $O$ is the center of the circumscribed circle of triangle $ABC$. Ray $AO$ intersects the side $BC$ at point $T$. With $AT$ as a diameter, a circle is constructed. At the intersection with the sides of the triangle $ABC$, three arcs were formed outside it. Prove that the larger of these arcs is equal to the sum of the other two. (Oleksii Karliuchenko)

Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$. Think six squares that has side length of $a_i$ ($i=1,2,\ldots,6$). Show that the squares can be packed inside a square of length $2$, without overlapping.

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal. [i]H. Lesov[/i]

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that $$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$

Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.