Kvanta MO 2018-21 VI-VIII (Ukraine) 6p

 geometry problems from Kvanta Mathematical Olympiad for grades 6-8 in Ukraine, with aops links in the names

2018 -2019,2021

it didn't take place in 2020

2018 1st Kvanta MO 7.4

In the triangle $ABC$, mark point $X$ on segment $BC$. Let $M$ be the midpoint of segment $AC$. The line $HM$ intersects the line passing through A parallel to $BC$ at point $E$. The segments $BE$ and $AC$ intersect at point $D$. Prove that if $BX = 2DM$, then $AD = AE$. 

(Arseniy Nikolaev)

2018 1st Kvanta MO 8.6

The $A$-excircle of the triangle $ABC$  touches the extension of the side $AB$ at the point $P$. Let $I$ be the incenter of the triangle, and $M$ be the midpoint of the segment $BI$. Prove that the points $P, M, B$ and $C$ lie on the same circle. 

(Anton Trygub)

2019 2nd Kvanta MO 7.6

In an acute-angled triangle $ABC$, the angle $BCA$ is $80^o$. On of the segment $BC$ there is such a point $X$ that $BX = AC$ and $\angle BAX = 45^o$ . Inside the triangle ABC is such a point $D$ that $\angle ABD= 40^o$ and $CD = AX$. Find the measure of the angle $BDC$.

(Arseniy Nikolaev)

2019 2nd Kvanta MO 8.3

Given a triangle $ABC$. A line parallel to $AC$ intersects the sides $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. The bisector of the angle $ABC$ intersects $A_1C_1$ at the point $K$. Let $X$ and $Z$ be the points of intersection of the lines $AK$ and $CK$ with the circumcircle of the triangle $ABC$, respectively (other than points $A$ and $C$). Prove that the quadrilateral $A_1XZC$ is cyclic. 

(Mykhailo Bondarenko)

2021 3rd Kvanta MO 7.6

A convex 35-gon is drawn on the board, in which all the diagonals are drawn (it's sides are also drawn). Rostislav and Kostya take turns finding an open polyline in which $1$ or $2$ links emerge from each vertex, and erase its edges (the polyline can self-intersect). The one who cannot make a move loses. Rostislav starts. Who wins when both play correctly?

(Arseniy Nikolaev)

2021 3rd Kvanta MO 8.5

Given a triangle $ABC$ with a right angle $A$ and an angle $CBA$ equal to $36^o$. Point $K$ of the hypotenuse $BC$ is such that $BK=AB-\frac{CB}{2}$. Find the angle $CKA$.

(Yana Kolodach)

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