Let $ABC$ be a triangle such that $AB<AC<BC$. Let $D,E$ be points on the segment $BC$ such that $BD=BA$ and $CE=CA$. If $K$ is the circumcenter of triangle $ADE$, $F$ is the intersection of lines $AD,KC$ and $G$ is the intersection of lines $AE,KB$, then prove that the circumcircle of triangle $KDE$ (let it be $c_1$), the circle with center the point $F$ and radius $FE$ (let it be $c_2$) and the circle with center $G$ and radius $GD$ (let it be $c_3$) concur on a point which lies on the line $AK$.

Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$for all possible values of $n$, as well as for which positive integers $n$ this minimum value is attained.

The positive real numbers $a,b,c,d$ satisfy the equality $$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$Find the maximum possible value of $a$.

Let $Q_n$ be the set of all $n$-tuples $x=(x_1,\ldots,x_n)$ with $x_i \in \{0,1,2 \}$, $i=1,2,\ldots,n$. A triple $(x,y,z)$ (where $x=(x_1,x_2,\ldots,x_n)$, $y=(y_1,y_2,\ldots,y_n)$, $z=(z_1,z_2,\ldots,z_n)$) of distinct elements of $Q_n$ is called a good triple, if there exists at least one $i \in \{1,2, \ldots, n \}$, for which $\{x_i,y_i,z_i \}=\{0,1,2 \}$. A subset $A$ of $Q_n$ will be called a good subset, if any three elements of $A$ form a good triple. Prove that every good subset of $Q_n$ contains at most $2 \cdot \left(\frac{3}{2}\right)^n$ elements.