AoPS Problem Making Contest 2016 +SL (APMC) 8p

geometry problems from AoPS Problem Making Contest 2016 (APMC 2016) and the Shortlist (unused problems) with aops links

2016

it took place only in 2016

contest announcement here

collected inside aops here

2016 AoPS problem making contest p1
Given triangle $ABC$ with the inner - bisector $AD$. The line passes through $D$ and perpendicular to $BC$ intersects the outer - bisector of $\angle BAC$ at $I$. Circle $(I,ID)$ intersects $CA$, $AB$ at $E$, $F$, reps. The symmedian line of $\triangle AEF$ intersects the circle $(AEF)$ at $X$. Prove that the circles $(BXC)$ and $(AEF)$ are tangent.

by baopbc 

2016 AoPS problem making contest p2

Let $ABC$ be a triangle with incenter $I$, and suppose that $AI$, $BI$, and $CI$ intersect $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let the circumcircles of $BDF$ and $CDE$ intersect at $D$ and $P$, and let $H$ be the orthocenter of $DEF$. Prove that $HI=HP$.

by ABCDE 

2016 AoPS problem making contest p4

Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.

by MouN 

2016 AoPS problem making contest p7

Let $\triangle ABC$ be given, it's $A-$mixtilinear incirlce, $\omega$, and it's excenter $I_A$. Let $H$ be the foot of altitude from $A$ to $BC$, $E$ midpoint of arc $BAC$ and denote by $M$ and $N$, midpoints of $BC$ and $AH$, respectively. Suposse that $MN\cap AE=\{ P \}$ and that line $I_AP$ meet $\omega$ at $S$ and $T$ in this order: $I_A-T-S-P$. Prove that circumcircle of $\triangle BSC$ and $\omega$ are tangent to each other.

by mihajlon

                                              2016 SL

2016 AoPS problem making contest SL G1

$\triangle ABC$ is given and it's circumcircle $\omega$. Let $X\in BC$ and let $Y\in \omega$ such that $AY\perp BC$. If perpedicular from $X$ to $BC$ meet $AB$ and $AC$ at $M$ and $N$, then prove that $XY$ and circumcircle of $\triangle AMN$ concur on $\omega$.

by aopser123 

2016 AoPS problem making contest SL G2
In circle $O$, $AB$ is a diameter. Construct points $C, D,$ and $F$ on circle $O$ such that $AC = AF$ and $ACDBF$ is a non-degenerate pentagon. Let $AD$ and $BC$ intersect at $E$. Line $DF$ intersects the circumcircle of $BOF$ again at $G$, and line $EG$ intersects this circumcircle again at $U$. Prove that $CD$ and $AB$ intersect at a point on the circumcircle of $AFU$.

by suli  

2016 AoPS problem making contest SL G3
Triangle $ABC (AB<BC)$ is incribed in a circle $(O, R)$ and let $M$ be the midpoint of $AC$. On segment $OB$ we take point $L$ such that $BL = \frac{{R + OM}}{2}$. The circle $(L, LB)$ intersects $AC$ and we name $K$ the point of intersection which is nearest to $A$. $BK$ inersects circle $(O)$ at $D$ and $DC$ intersects circle $(C, a-c)$ at $E$. Prove that $AD=DE$.

by george_54

2016 AoPS problem making contest SL G4
Let $ABCD$ be a cyclic quadrilateral such that dioganals $AC$ and $BD$ intersect at $P$.Circumcircles of triangles $ADP$ and $BCP$ intersect at $Q$.Let $M$ and $N$ be the midpoints of $AD$ and $BC$ respectively.Circumcircle of $MQN$ intersects $\odot ADP$ and $\odot BCP$ at $X$ and $Y$ respectively.Prove that $XN$ and $YM$ intersects at $P$.

by Anar Abbas (Analgin)