Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

An L-shape is one of the following four pieces, each consisting of three unit squares: [asy][asy] size(300); defaultpen(linewidth(0.8)); path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1); draw(P); draw(shift((2.7,0))*rotate(90,(1,1))*P); draw(shift((5.4,0))*rotate(180,(1,1))*P); draw(shift((8.1,0))*rotate(270,(1,1))*P); [/asy][/asy] A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares. We say that a placement of L-shapes on unmarked unit squares is called $\textit{good}$ if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board. B wins if every $\textit{good}$ placement of L-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which B has a winning strategy.