Problems

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle $CDE$ the angle $\mathrm{\angle }CDE={90}^{\circ }$ and the line $DH$ is the median. A circle with center $A$ is inscribed in the triangle $CDH$ and is tangent to the segment $DH$ in its middle, let’s denote it as $G$, so $GH=DG$. Find the angles of the triangle $CDE$.

A circle with center $A$ is inscribed into a square $CDFE$. A line $GH$ intersects the sides $CD$ and $CE$ of the square and is tangent to the circle at the point $I$. Find the perimeter of the triangle $CHG$ (the sum of lengths of all the sides) if the side of the square is $10$cm.

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle $EFG$ the line $EH$ is the median. Two circles with centres $A$ and $C$ are inscribed into triangles $EFH$ and $EGH$ respectively, they are tangent to the median $EH$ at the points $B$ and $D$. Find the length of $BD$ if $EF-EG=2$.

Is it possible to cover a $6\times 6$ board with the $L$-tetraminos without overlapping? The pieces can be flipped and turned.

Is it possible to cover a $(4n+2)\times (4n+2)$ board with the $L$-tetraminos without overlapping for any $n$? The pieces can be flipped and turned.

Is it possible to cover a $4n\times 4n$ board with the $L$-tetraminos without overlapping for any $n$? The pieces can be flipped and turned.

Each number denotes the area of a rectangle it is written into. What is the area of the last rectangle?

Divide the trapezium into two parts such that they can be reassembled to make a triangle

Which of the triangles has the smallest area?

In a square $ABHI$ two smaller squares are drawn: $ACFG$ with area equal to $16$ and $BCED$ with area equal to $4$. Find the area of hexagon $DEFGIH$.