Oliforum Contest I-V 2008-17 (Italy) 9p

geometry problems from an contest from the Italian forum Oliforum.it with aops links in the names

aops post collection

geometry collected inside aops here

2008-09, 2012-13, 2017

2008 Oliforum Contest I 1.2 

Let $ABCD$ be a cyclic quadrilateral with $AB>CD$ and $BC>AD$. Take points $X$ and $Y$ on the sides $AB$ and $BC$, respectively, so that $AX=CD$ and $AD=CY$. Let $M$ be the midpoint of $XY$. Prove that $AMC$ is a right angle.

2008 Oliforum Contest I 2.3 

Let $C_1,C_2$ and $C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $i,j$, we define $(r_{ij},s_{ij})$ as the two internal tangent lines of $(C_i,C_j)$. Let $r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ABCA'B'C'$.

Prove that $AA',BB'$ and $CC'$ are concurrent.

2009 Oliforum Contest II 1.3 

Let a cyclic quadrilateral $ABCD$, $AC \cap BD = E$ and let a circle $\Gamma$ internally tangent to the arch $BC$ (that not contain $D$) in $T$ and tangent to $BE$ and $CE$. Call $R$ the point where the angle bisector of $\angle ABC$ meet the angle bisector of $\angle BCD$ and $S$ the incenter of $BCE$. Prove that $R$, $S$ and $T$ are collinear.

(Gabriel Giorgieri)

2009 Oliforum Contest II final p2 

Let a convex quadrilateral $ABCD$ fixed such that $AB = BC$, $\angle ABC = 80, \angle CDA = 50$. Define $E$ the midpoint of $AC$; show that $\angle CDE = \angle BDA$

(Paolo Leonetti)

2012 Oliforum Contest III p5

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

2013 Oliforum Contest IV p2

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

2013 Oliforum Contest IV p6

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.

2017 Oliforum Contest V p2

Find all quadrilaterals which can be covered (without overlappings) with squares with side $1$ and equilateral triangles with side $1$.

(Emanuele Tron)

2017 Oliforum Contest V p9

Given a triangle $ABC$, let $P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $P$ on the sides of the triangle ABC. Show that $P$ is the barycenter of $DEF$.

(Jack D’Aurizio)