LuMaT 2018-19 6p (Indonesia)

geometry problems from Indonesian LuMaT with aops links in the names

Lomba Unik Matematika ala Tobi

Finals 2018 - 2019 SMA (Seniors)

2018 Indonesia LuMaT SMA Finals 1.3

Given an isosceles acute triangle $ABC$ ($AB = AC$) with circumcircle $\Gamma$ and point $P,Q$ are chosen on segment $BC$ ($P$ is closer to $B$ than $Q$). Line $AP$ and $AQ$ intersect $\Gamma$ again at $P_0, Q_0$. Points $P_1, Q_1$ are respectively midpoints of arcs $AP_0, BQ_0$ that contain $B,C$. Rays $AP_1$ and $AQ_1$ intersects $BC$ at $C_1$ and $B_1$. Prove that if the midpoint of $BB_1$ lies on the incircle of $\triangle ACQ$, then the midpoint of $CC_1$ also lies on the incircle of $\triangle ABP$. 

2018 Indonesia LuMaT SMA Finals 2.5

Suppose $P$ is an inner point in a convex pentagon $ABCDE$  such that $$ \angle APB = \angle BPC = \angle CPD = \angle DPE = \angle EPA$$ and all 5 $P$-excircles of $\triangle APB, \triangle BPC, \triangle CPD, \triangle DPE, \triangle EPA$ have the same radius. Prove that $ABCDE$ is a regular polygon.

2019 Indonesia LuMaT SMA Finals 1.4

Given two distinct points $A,B$ at the plane. For any other point $X$, define $X_B \not= B$ as the intersection of both line $XB$ and the circle that passes through $A$ and $B$ and tangent to $XA$. Suppose that $P,Q$ are points at the plane such that $PQ$ intersects $AB$ and $P_B, Q_B$ is well defined. Prove that $\angle PAQ = 90^{\circ}$ if and only if the tangents of $P_B, Q_B$ with respect to the circumcircle of $P_B BQ_B$  intersect at line $PQ$. 

2019 Indonesia LuMaT SMA Finals 2.6

Given that $H$ is the orthocenter of an acute triangle $ABC$, and $AH$ intersects $(ABC)$ at $D$. The tangent to $(BCH)$ that passes through $H$ intersects $AB$ and $AC$ at $E$ and $F$. Prove that $(DEF)$ is tangent to $(ABC)$.

Finals 2019 - 2020 SMP (Juniors)

2019 Indonesia LuMaT Final SMP  p3 

Mrs Ippi will fish in a triangular pond of side length 6,7,8 with a fishing pole of length t. What is the mininmal value of t, so that wherever the fish inside the pool is located, Mrs. Ippi can throw the bait from outside of the point on top of the fish?

2020 Indonesia LuMaT Final SMP  p3 

On a triangle ABC, points P,Q on the segment BC ( and B,P,Q,C are four different points) are given such that BP=CQ and \angle BAP = \angle QAC. Prove that ABC is isosceles