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KöMaL A

geometry problems with aops links from 

KöMaL , 
Mathematical and Physical Journal for High Schools
  Hungarian Magazine Magazine 

A 701-750

KöMaL A.702
Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $ \angle A=\angle X, \angle B=\angle Y,  \angle C=\angle Z$). Of all the elegant triangles, which one has the smallest perimeter?

KöMaL A.705
Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P!=B$, and circle $CHD$ meets $AC$ at $Q!=C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

KöMaL A. 716
Let $ABC$ be a triangle and let $D$ be a point in the interior of the triangle which lies on the angle bisector of $\angle$$BAC$. Suppose that lines $BD$ and $AC$ meet at $E$, and that lines $CD$ and $AB$ meet at $F$. The circumcircle of $ABC$ intersects line $EF$ at points $P$ and $Q$. Show that if $O$ is the circumcenter of $DPQ$, then $OD$ is perpendicular to $BC$.

KöMaL A. 719
Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1, B_1$, and $C1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel.

KöMaL A. 724
A sphere $G$ lies within tetrahedron $ABCD$, touching faces $ABD$, $ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $G$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $G$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.

In triangle $ABC$ with incenter $I$, line $AI$ intersects the circumcircle of $ABC$ at $S\ne A$. Let the reflection of $I$  with respect to $BC$ be $J$, and suppose that line $SJ$ intersects the circumcircle of $ABC$ for the second time at point $P\ne S$. Show that $AI=PI.$

In a cyclic quadrilateral $ABCD$, the diagonals meet at point $E$, the midpoint of side $AB$ is $F$, and the feet of perpendiculars from $E$ to the lines $DA$, $AB$, and $BC$ are $P$, $Q$, and $R$, respectively. Prove that the points $P, Q, R$, and $F$ are concyclic.

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A \ne X$ and $B \ne X$. Prove that the lines $AB$are either all tangent to a fixed circle, or they all pass through a point.

Let $P$ be a point in the plane of triangle $ABC$. Denote the reflections of $A, B, C$ about $P$ by $A'$, $B'$ and $C'$, respectively. Let $A''$,$ B''$,$ C''$ be the reflections of $A'$, $B'$, $C'$ over the lines $BC, CA$ and $AB$ respectively. Let the line $A'' B''$ intersect $AC$ at $A_b$ and let $A''C''$ intersect $AB$ at a point $A_c$. Denote by $\omega_A$ the circle through the points $A, Ab, Ac$. The circles $\omega_B$, $\omega_C$ are defined similarly. Prove that $\omega_A$ , $\omega_B$, $omega_C$ are coaxial, i.e., they share a common radical axis.

Convex quadrilateral $ABCD$ is inscribed in circle $\Omega$. Its sides $AD$ and $BC$ intersect at point $E$. Let $M$ and $N$ be the midpoints of the circle arcs $AB$ and $CD$ not containing the other vertices, and let $I, J, K, L$ denote the incentres of triangles $ABD, ABC, BCD, CDA$, respectively. Suppose $\Omega$ intersects circles $IJM$ and $KLN$ for the second time at points $U!=M$ and $V!=N$. Show that the points $E, U$, and $V$ are collinear.

The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ ($BP<BQ$). Let $UV$ be the diameter of the incircle perpendicular to $AC$ ($BU \le BV$). Show that the lines $AC, PV$ and $QU$ pass through one point.

The circles $\Omega$ and $\omega$ in its interior are fixed. The distinct points $A, B, C, D, E$ move on $\Omega$ in such a way that the line segments $AB, BC, CD$ and $DE$ are tangents to $\omega$. The lines $AB$ and $CD$ meet at point $P$, the lines $BC$ and $DE$ meet at $Q$. Let $R$ be the second intersection of the circles $BCP$ and $CDQ$, other than $C$. Show that $R$ moves either on a circle or on a line.

Let $k1,…,k5$ be five circles in the plane such that $k_1$ and $k_2$ are externally tangent to each other at point $T$,$k_3$ and $k_4$ are externally tangent to both $k_1$ and $k_2$, $k_5$ is externally tangent to $k_3$ and $k_4$ at points $U$ and $V$, respectively, moreover $k_5$ intersects $k1$ at $P$ and $Q$.Show that $(PU*PV)/(QU*QV)=PT^2/QT^2.$

source: https://www.komal.hu/verseny/korabbi.e.shtml


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