Find the least number of buttons that can be placed on the squares of a $5 \times 5$ grid so that no two buttons are on the same square or on squares with a common side (buttons may be on squares with a common vertex) and no buttons can be added to the grid under the same conditions.

Olesya wrote down a natural number $N$. After this, Andriy wrote down one sixth, one fifth, one fourth, one third and a half of $N$. It turns out that the sum of all numbers that are written is integer. What is the least possible number that Olesya could write?

Let $\triangle ABC$ be a triangle with $AB < AC$, and let $D$ be a point on the side $AC$ such that $AD = \frac{AC-AB}{2}$. Points $X$ and $Y$ are chosen on the line through $A$ parallel to $BC$ such that $BX = CY$ and line $AC$ is tangent to the circumcircle of $\triangle XDY$. Prove that the tangents to the circumcircle of $\triangle XDY$ at points $X$ and $Y$ meet on line $BC$.

Are there any 10 numbers, not all of which are the same, each of which is equal to the square of the sum of all the other numbers?