IGO 2017

In the regular pentagon $ABCDE$ , the perpendicular at $C$ to $CD$ meets $AB$ at $F$ . Prove that $AE+AF=BE$ .

Proposed by Alireza Cheraghi

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle clockwise if the increasing order of them is in clockwise order. Can the number of clockwise triangles be exactly $2017$ ?

Proposed by Morteza Saghafian

Let $ABC$ be an acute-angled triangle with $A=60^{\circ}$ . Let $E,F$ be the feet of altitudes through $B,C$ respectively. Prove that $CE-BF=\tfrac{3}{2}(AC-AB)$ Proposed by Fatemeh Sajadi

Two circles $\omega_1,\omega_2$ intersect at $A,B$ . An arbitrary line through $B$ meets $\omega_1,\omega_2$ at $C,D$ respectively. The points $E,F$ are chosen on $\omega_1,\omega_2$ respectively so that $CE=CB,\ BD=DF$ . Suppose that $BF$ meets $\omega_1$ at $P$ , and $BE$ meets $\omega_2$ at $Q$ . Prove that $A,P,Q$ are collinear.

Proposed by Iman Maghsoudi

On the plane, $n$ points are given ( $n>2$ ). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given $n$ ?

Proposed by Boris Frenkin (Russia)

Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ( $X$ is between $B,Y$ ). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$ . Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$ , and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$ . Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$ .

Proposed by Iman Maghsoudi

In triangle $ABC$ , the incircle, with center $I$ , touches the sides $BC$ at point $D$ . Line $DI$ meets $AC$ at $X$ . The tangent line from $X$ to the incircle (different from $AC$ ) intersects $AB$ at $Y$ . If $YI$ and $BC$ intersect at point $Z$ , prove that $AB=BZ$ .

Proposed by Hooman Fattahimoghaddam

We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.

Proposed by Mohammad Ali Abam - Morteza Saghafian

Let $O$ be the circumcenter of triangle $ABC$ . Line $CO$ intersects the altitude from $A$ at point $K$ . Let $P,M$ be the midpoints of $AK$ , $AC$ respectively. If $PO$ intersects $BC$ at $Y$ , and the circumcircle of triangle $BCM$ meets $AB$ at $X$ , prove that $BXOY$ is cyclic.

Proposed by Ali Daeinabi - Hamid Pardazi

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$ . Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$ .

Proposed by Alexey Zaslavsky (Russia)