VU MIF Olympiad 2016-22 (Lithuania) 12p

geometry problems from VU MIF Olympiad (Vilnius University, Lithuania)  with aops links in the names

started in 2016,

collected inside aops here

2016-19, 2022

did not take place in 2020, 2021

2016 VU MIF Olympiad IX p2

$ABC$ is an isosceles triangle ($AB = AC$) On side $AB$ lie points $K$ and $M$ and on side $AC$ lie point $L$, such that $BC =CM = ML = LK = KA$. Find the angles of triangle $ABC$.

2016 VU MIF Olympiad X p2

Two circles with no common points are given and their common external tangents are drawn. One of them touches the first circle at point $A$ and the other touches the second at point $D$. The line $AD$ again intersects the first circle at point $B$ and the second at point $C$. Prove that $AB \parallel CD$.

2016 VU MIF Olympiad XI XII p2

The quadrilateral $ABCD$ is inscribed in a circle. In the segment $AB$ there is a point $Q$ such that $AQ = CD$, and in the segment $AD$, the point $P$ such that $AP = BC$. In what ratio is the segment  $AC$ divided by segment $PQ$?

2017 VU MIF Olympiad IX p3 

The segments $AD, BE, CF$ are the angle bisectors of triangle $ABC$ with angle $\angle A=120^o$ . Find $\angle EDF$.

2017 VU MIF Olympiad X, XI, XII p3

The point $L$ is marked in the hypotenuse $AB$ of the right triangle $ABC$. The circumcircle of the triangle $ACL$  intersects line $BC$ at a point $M \ne C$, and the circumcircle of the triangle  $BCL$ intersects line $AC$ at point $N \ne C$. Find the angle between the lines $AM$ and $BN$.

2018 VU MIF Olympiad IX p3

The vertex $C$ of the square $ABCD$ belongs to the line passing through the side $EF$ of the rhombus $BDEF$ (point $F$ is between points $E$ and $C$). Find angle $CBF$.

2018 VU MIF Olympiad X p3

Point $P$ is marked on the outside of the right triangle $ABC$ ($\angle ACB = 90^o$ )  such that the points $C$ and $P$ are on one side of the line $AB$, and segments $AB$ and $PB$ are equal and perpendicular. On the line $PC$ is marked a point $D$ such that the lines $PC$ and $AD$ are perpendicular and on the line $AD$ the point $E$ is marked such that the lines $BC$ and $BE$ are perpendicular. Find the angle $BCE$

2018 VU MIF Olympiad XI, XII p3

The altitudes of the acute triangle $ABC$ intersect at point $H$. Point $M$ divides the side $BC$ in half. The point $P$ is marked on the segment $AM$  such  that the segments $AM$ and $HP$ are perpendicular. Prove that $MA \cdot  MP = MB^2$

2019 VU MIF Olympiad IX, X p2 (part a for IX)

Square $ABCX$, equilateral triangle $ABC'$ , regular pentagon $ABDY_1Y_2$, regular $n$-gon $ABD'Z_1Z_2... Z_{n -3}$ and regular $1000$-gon $ABEU_1U_2...U_{997}$ have a common side $AB$. Points $D$ and $D'$ lie inside the $1000$-gon and the points $C$ and $C'$ lie on the outside. Find :

a) $\angle CED$ ,

b) $n> 3$ such  that  $\angle C' ED' =  \angle CED + 1^o$.

2019 VU MIF Olympiad XI, XII p3

The point $C \ne A, B$ belongs to a circle of diameter $AB$. Point $D$ divides the shorter arc $BC$ in  half. Lines $AD$ and $BC$ intersects at point $E$. Find the length of the segment $AB$ if $CE = 3$, $BD = 2\sqrt5$ .

2022 VU MIF Olympiad IX p3

Trapezoid $ABCD$ with bases $BC$ and $AD$ and lateral sides $AB$ and $CD$, their midpoints are denoted $K, L, M, N$, respectively. Find the lengths of the bases $BC$ and $AD$, if $\angle DAB = 70^o$, $\angle CDA = 20^o$, $KL = 2$, $MN = 4$.

2022 VU MIF Olympiad XI, XII p3

On the sides $AB$ and $AC$ of the triangle $ABC$, the points $M$ and $N$ are marked, respectively that $MC = AC$, $NB = AB$. The points $A$ and $P$ are symmetric wrt the line $BC$ . Prove that the line PA bisects the angle $MPN$.

source: http://mif.vu.lt/matematikos-olimpiados/mif/