IGO 2019

As shown in the figure, there are two rectangles $ABCD$ and $PQRD$ with the same area, and with parallel corresponding edges. Let points $N,$ $M$ and $T$ be the midpoints of segments $QR,$ $PC$ and $AB$ , respectively. Prove that points $N,M$ and $T$ lie on the same line.

Proposed by Morteza Saghafian

There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all?

Proposed by Boris Frenkin - Russia

Quadrilateral $ABCD$ is given such that $$\angle DAC = \angle CAB = 60^\circ,$$ and $$AB = BD - AC.$$ Lines $AB$ and $CD$ intersect each other at point $E$ . Prove that $$\angle ADB = 2\angle BEC.$$

Proposed by Iman Maghsoudi

For a convex polygon (i.e. all angles less than $180^\circ$ ) call a diagonal bisector if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?

Proposed by Morteza Saghafian

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$ , and point $O_1$ lies on $\omega_2$ . Let $P$ be an arbitrary point lying on $\omega_1$ . Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$ , $Y$ and $C$ , respectively. Prove that quadrilateral $XPYC$ is a parallelogram.

Proposed by Iman Maghsoudi

Find all quadrilaterals $ABCD$ such that all four triangles $DAB$ , $CDA$ , $BCD$ and $ABC$ are similar to one-another.

Proposed by Morteza Saghafian

Three circles $\omega_1$ , $\omega_2$ and $\omega_3$ pass through one common point, say $P$ . The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$ , respectively. Points $P_{2,1}$ , $P_{2,3}$ , $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$ , $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.

Proposed by Mahdi Etesamifard

Let $ABCD$ be a parallelogram and let $K$ be a point on line $AD$ such that $BK=AB$ . Suppose that $P$ is an arbitrary point on $AB$ , and the perpendicular bisector of $PC$ intersects the circumcircle of triangle $APD$ at points $X$ , $Y$ . Prove that the circumcircle of triangle $ABK$ passes through the orthocenter of triangle $AXY$ .

Proposed by Iman Maghsoudi

Let $ABC$ be a triangle with $\angle A = 60^\circ$ . Points $E$ and $F$ are the foot of angle bisectors of vertices $B$ and $C$ respectively. Points $P$ and $Q$ are considered such that quadrilaterals $BFPE$ and $CEQF$ are parallelograms. Prove that $\angle PAQ > 150^\circ$ . (Consider the angle $PAQ$ that does not contain side $AB$ of the triangle.)

Proposed by Alireza Dadgarnia

Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$ . Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that $\angle ABC = 90^\circ$ . Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$ . Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$ . Prove that points $X, Y$ and $Z$ are collinear.

Proposed by Iman Maghsoudi

Is it true that in any convex $n$ -gon with $n > 3$ , there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute?

Proposed by Boris Frenkin - Russia

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$ , respectively. These two circles intersect at points $X$ and $Y$ . $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$ . Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$ , respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$ . Prove that lines $AM, BN$ and $O_1O_2$ concur.

Proposed by Dominik Burek - Poland

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$ . $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$ ). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$ . Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$ . Let $\omega$ be the circumcircle of triangle $ZMN$ . Line $AM$ meets $\omega$ for the second time at $P$ . Let $K$ be a point on $\omega$ such that $KN\parallel AM$ , $\omega_b$ be a circle that passes through $B$ , $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$ , $Y$ and tangents to $BC$ . Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$ , $\omega_c$ and $\Gamma$ .

Proposed by Tran Quan - Vietnam

Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$ , orthocenter of triangle $ABC$ , coincides with the focus of parabola $\Delta$ . Prove that by changing the position of points $A, B$ and $C$ on $\Delta$ so that the orthocenter remain at $H$ , inradius of triangle $ABC$ remains unchanged.

Proposed by Mahdi Etesamifard