In an acute-angled triangle $ABC$, the circle with diameter $[AB]$ intersects the altitude drawn from vertex $C$ at a point $D$ and the circle with diameter $[AC]$ intersects the altitude drawn from vertex $B$ at a point $E$. Let the lines $BD$ and $CE$ intersect at $F$. Prove that $$AF\perp DE$$

For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$is an integer?

In a country, there are $28$ cities and between some cities there are two-way flights. In every city there is exactly one airport and this airport is either small or medium or big. For every route which contains more than two cities, doesn't contain a city twice and ends where it begins; has all types of airports. What is the maximum number of flights in this country?

Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$Find the maximum value of the expression $$x^3+2y$$

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ lucky. For which positive integers $n$, one can find a lucky $n$-tuple?

Initially on a blackboard, the equation $a_1x^2+b_1x+c=0$ is written where $a_1, b_1, c_1$ are integers and $(a_1+c_1)b_1 > 0$. At each move, if the equation $ax^2+bx+c=0$ is written on the board and there is a $x \in \mathbb{R}$ satisfying the equation, Alice turns this equation into $(b+c)x^2+(c+a)x+(a+b)=0$. Prove that Alice will stop after a finite number of moves.

$w_1$ and $w_2$ circles have different diameters and externally tangent to each other at $X$. Points $A$ and $B$ are on $w_1$, points $C$ and $D$ are on $w_2$ such that $AC$ and $BD$ are common tangent lines of these two circles. $CX$ intersects $AB$ at $E$ and $w_1$ at $F$ second time. $(EFB)$ intersects $AF$ at $G$ second time. If $AX \cap CD =H$, show that points $E, G, H$ are collinear.