Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

Let $a,b,c\in (0,1). $ Prove that $$a+b+c+2abc > ab+bc+ca+2\sqrt{abc}.$$JBMO TST Serbia 2019

$3.$ Congruent circles $k_{1}$ and $k_{2}$ intersect in the points $A$ and $B$. Let $P$ be a variable point of arc $AB$ of circle $k_{2}$ which is inside $k_{1}$ and let $AP$ intersect $k_{1}$ once more in point $C$, and the ray $CB$ intersects $k_{2}$ once more in $D$. Let the angle bisector of $\angle CAD$ intersect $k_{1}$ in $E$, and the circle $k_{2}$ in $F$. Ray $FB$ intersects $k_{1}$ in $Q$. If $X$ is one of the intersection points of circumscribed circles of triangles $CDP$ and $EQF$, prove that the triangle $CFX$ is equilateral.

$4.$ On a table there are notes of values: $1$, $2$, $5$, $10$, $20$ ,$50$, $100$, $200$, $500$, $1000$, $2000$ and $5000$ (the number of any of these notes can be any non-negative integer). Two players , First and Second play a game in turns (First plays first). With one move a player can take any one note of value higher than $1$ , and replace it with notes of less value. The value of the chosen note is equal to the sum of the values of the replaced notes. The loser is the player which can not play any more moves. Which player has the winning strategy?