Malaysia Juniors SHL 2015 8p

  geometry problems from Junior Olympiad of Malaysia  with aops links in the names

2013, 2015

2013 Malaysia Juniors problem 2

Consider a triangle $ABC$ with height $AH$ and $H$ on $BC$. Let $\gamma_1$ and $\gamma_2$ be the circles with diameter $BH,CH$ respectively, and let their centers be $O_1$ and $O_2$. Points $X,Y$ lie on $\gamma_1,\gamma_2$ respectively such that $AX,AY$ are tangent to each circle and $X,Y,H$ are all distinct. $P$ is a point such that $PO_1$ is perpendicular to $BX$ and $PO_2$ is perpendicular to $CY$.

Prove that the circumcircles of $PXY$ and $AO_1O_2$ are tangent to each other.

2015 Malaysia Juniors Shortlist G5 problem 2

Let $ABCD$ be a convex quadrilateral. Let angle bisectors of $\angle B$ and $\angle C$ intersect at $E$. Let $AB$ intersect $CD$ at $F$. Prove that if $AB+CD=BC$, then $A,D,E,F$ are concyclic.

2015 Shortlist

2015 Malaysia Juniors Shortlist G1

Given a triangle $ABC$, and let $E$ and $F$ be the feet of altitudes from vertices $B$ and $C$ to the opposite sides. Denote $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$. Given that $FA=FC$, prove that $OEHF$ is a parallelogram.

2015 Malaysia Juniors Shortlist G2

Let $ABC$ be a triangle, and let $M$ be midpoint of $BC$. Let $I_b$ and $I_c$ be incenters of $AMB$ and $AMC$. Prove that the second intersection of circumcircles of $ABI_b$ and $ACI_c$ distinct from $A$ lies on line $AM$.

2015 Malaysia Juniors Shortlist G3

Let $ABC$ a triangle. Let $D$ on $AB$ and $E$ on $AC$ such that $DE||BC$. Let line $DE$ intersect circumcircle of $ABC$ at two distinct points $F$ and $G$ so that line segments $BF$ and $CG$ intersect at P. Let circumcircle of $GDP$ and $FEP$ intersect again at $Q$. Prove that $A, P, Q$ are collinear.

2015 Malaysia Juniors Shortlist G4

Let $ABC$ be a triangle and let $AD, BE, CF$ be cevians of the triangle which are concurrent at $G$. Prove that if $CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE$ then $AG \le GD$.

2015 Malaysia Juniors Shortlist G5 problem 2

Let $ABCD$ be a convex quadrilateral. Let angle bisectors of $\angle B$ and $\angle C$ intersect at $E$. Let $AB$ intersect $CD$ at $F$. Prove that if $AB+CD=BC$, then $A,D,E,F$ are concyclic.

2015 Malaysia Juniors Shortlist G6

Let $ABC$ be a triangle. Let $\omega_1$ be circle tangent to $BC$ at $B$ and passes through $A$. Let $\omega_2$ be circle tangent to $BC$ at $C$ and passes through $A$. Let $\omega_1$ and $\omega_2$ intersect again at $P \neq A$. Let $\omega_1$ intersect $AC$ again at $E\neq A$, and let $\omega_2$ intersect $AB$ again at $F\neq A$. Let $R$ be the reflection of $A$ about $BC$, Prove that lines $BE, CF, PR$ are concurrent.

2015 Malaysia Juniors Shortlist G7

Let $ABC$ be an acute triangle. Let $H_A,H_B,H_C$ be points on $BC,AC,AB$ respectively such that $AH_A\perp BC, BH_B\perp AC, CH_C\perp AB$. Let the circumcircles $AH_BH_C,BH_AH_C,CH_AH_B$ be $\omega_A,\omega_B,\omega_C$ with circumcenters $O_A,O_B,O_C$ respectively and define $O_AB\cap \omega_B=P_{AB}\neq B$. Define $P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}$ similarly. Define circles $\omega_{AB},\omega_{AC}$ to be $O_AP_{AB}H_C,O_AP_{AC}H_B$ respectively. Define circles $\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}$ similarly.

Prove that there are $6$ pairs of tangent circles in the $6$ circles of the form $\omega_{xy}$.

2015 Malaysia Juniors Shortlist G8

Let $ABCDE$ be a convex pentagon such that $BC$ and $DE$ are tangent to the circumcircle of $ACD$. Prove that if the circumcircles of $ABC$ and $ADE$ intersect at the midpoint of $CD$, then the circumcircles $ABE$ and $ACD$ are tangent to each other.