Matol Online Olympiad 2021-22 (Kazakshtan) 8p

 geometry problems from  Matol Online Olympiad (Kazakshtan) with aops links in the names

2020-21/ 2021-22

1st Matol Online Olympiad 2020-21 7.2 8.2

In a convex quadrilateral $ABCD$, $\angle A = \angle D < 90^o$ , $\angle C = 90^o$. It turned out that the distance from point $A$ to line $CD$ is equal to segment $CD$. Find the degree measure of the angle $ADB$. 

1st Matol Online Olympiad 2020-21 9.4

In a convex quadrilateral $ABCD$, $\angle A = \angle D < 90^o$ , $\angle C = 90^o$. It turned out that the distance from point $A$ to line $CD$ is equal to segment $CD$. Find the degree measure of the angle $ADB$. 

1st Matol Online Olympiad 2020-21 10.3 11.3

In a convex quadrilateral $ABCD$, $\angle A = \angle D < 90^o$ , $\angle C = 90^o$. It turned out that the distance from point $A$ to line $CD$ is equal to segment $CD$. Find the degree measure of the angle $ADB$. 

2nd Matol Online Olympiad 2021-22 7.3

In the square $ABCD$ on the side $BC$ and $CD$ mark the points $M$ and $K$, respectively, such that $\angle BAM=\angle CKM=30^o$. Find $\angle MKA$ .

2nd Matol Online Olympiad 2021-22 8.2

The convex quadrilateral $ABCD$ satisfies $BC=AD$, $AB=AC$, $\angle BCD=90^o$. Find all the possible values of angle $\angle ADC$.

2nd Matol Online Olympiad 2021-22 9.2

In the quadrilateral $ABCD$ on the side $BC$, mark points $N,M$ such that $BN=NM=MC$, and on the side $AD$ mark points $K,L$ such that $AK=KL=LD$ . Prove that $AB+CD \ge KN + LM$.

2nd Matol Online Olympiad 2021-22 10.3

The incircle and the excircle of a triangle $ABC$, with $\angle C=90^o$, touch the side $BC$ at points $A_1$ and $A_2$, respectively. In a similar way, we define the points $B_1$ and $B_2$ . Find the acute angle between lines $A_1B_2$ and $B_1A_2$.

2nd Matol Online Olympiad 2021-22 11.2

Distance between the midpoints of the trapezoid diagonals $ABCD$ is equal to $a$. It is known that a certain circle touches diagonals $AC$ and $BD$, as well as extensions of its bases $AD$ and $BC$ at points $D$ and $C$, respectively. Find the absolute of the difference in the lengths of the diagonals.

source: http://matol.kz/