Individual
On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.
Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers
We have $10$ identical tiles as shown. The tiles can be rotated, but not flipper over. A $7 \times 7$ board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit sqaures that can be covered with two tiles.
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.
A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?
Team
The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?
Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.
Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.
Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.
There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.