geometry problems from Mathematics Regional Olympiad of Mexico Southeast with aops links in the names
In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.
In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.
Let $ABCD$ a trapezium with $AB$ parallel to $CD, \Omega$ the circumcircle of $ABCD$ and $A_1,B_1$ points on segments $AC$ and $BC$ respectively, such that $DA_1B_1C$ is a cyclic cuadrilateral. Let $A_2$ and $B_2$ the symmetric points of $A_1$ and $B_1$ with respect of the midpoint of $AC$ and $BC$, respectively. Prove that points $A, B, A_2, B_2$ are concyclic.
The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that $$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$ then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.
Let $M$ the midpoint of $AC$ of an acutangle triangle $ABC$ with $AB>BC$. Let $\Omega$ the circumcircle of $ABC$. Let $P$ the intersection of the tangents to $\Omega$ in point $A$ and $C$ and $S$ the intersection of $BP$ and $AC$. Let $AD$ the altitude of triangle $ABP$ with $D$ in $BP$ and $\omega$ the circumcircle of triangle $CSD$. Let $K$ and $C$ the intersections of $\omega$ and $\Omega (K\neq C)$. Prove that $\angle CKM=90^\circ$
Let $ABC$ a triangle and $C$ it´s circuncircle. Let $D$ a point in arc $AB$ that not contain $A$, diferent of $B$ and $C$ such that $CD$ and $AB$ are not parallel. Let $E$ the intersection of $CD$ and $AB$ and $O$ the circumcircle of triangle $DBE$. Prove that the measure of $\angle OBE$ does not depend of the choice of $D$.
Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.
Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$
Let $ABC$ an isosceles triangle with $CA=CB$ and $\Gamma$ it´s circumcircle. The perpendicular to $CB$ through $B$ intersect $\Gamma$ in points $B$ and $E$. The parallel to $BC$ through $A$ intersect $\Gamma$ in points $A$ and $D$. Let $F$ the intersection of $ED$ and $BC, I$ the intersection of $BD$ and $EC, \Omega$ the cricumcircle of the triangle $ADI$ and $\Phi$ the circumcircle of $BEF$.If $O$ and $P$ are the centers of $\Gamma$ and $\Phi$, respectively, prove that $OP$ is tangent to $\Omega$.
Let $ABCD$ a convex quadrilateral. Suppose that the circumference with center $B$ and radius $BC$ is tangent to $AD$ in $F$ and the circumference with center $A$ and radius $AD$ is tangent to $BC$ in $E$. Prove that $DE$ and $CF$ are perpendicular.
Let $\Gamma$ a circumference. $T$ a point in $\Gamma$, $P$ and $A$ two points outside $\Gamma$ such that $PT$ is tangent to $\Gamma$ and $PA=PT$. Let $C$ a point in $\Gamma (C\neq T)$, $AC$ and $PC$ intersect again $\Gamma$ in $D$ and $B$, respectively. $AB$ intersect $\Gamma$ in $E$. Prove that $DE$ it´s parallel to $AP$.
Let $ABC$ a triangle with $AB<AC$ and let $I$ it´s incenter. Let $\Gamma$ the circumcircle of $\triangle BIC$. $AI$ intersect $\Gamma$ again in $P$. Let $Q$ a point in side $AC$ such that $AB=AQ$ and let $R$ a point in $AB$ with $B$ between $A$ and $R$ such that $AR=AC$. Prove that $IQPR$ is cyclic.
Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.
Let $A, B$ and $C$ three points on a line $l$, in that order .Let $D$ a point outside $l$ and $\Gamma$ the circumcircle of $\triangle BCD$, the tangents from $A$ to $\Gamma$ touch $\Gamma$ on $S$ and $T$. Let $P$ the intersection of $ST$ and $AC$. Prove that $P$ does not depend of the choice of $D$.
No comments:
Post a Comment