Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

Consider $ABCD$ a square of side $1$. Points $P,Q,R,S$ are chosen on sides $AB$, $BC$, $CD$ and $DA$ respectively such that $|AP| = |BQ| =|CR| =|DS| = a$, with $a < 1$. The segments $AQ$, $BR$, $CS$ and $DP$ are drawn. Calculate the area of the quadrilateral that is formed in the center of the figure. [asy][asy] unitsize(1 cm); pair A, B, C, D, P, Q, R, S; A = (0,3); B = (0,0); C = (3,0); D = (3,3); P = (0,2); Q = (1,0); R = (3,1); S = (2,3); draw(A--B--C--D--cycle); draw(A--Q); draw(B--R); draw(C--S); draw(D--P); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE); label("$D$", D, NE); label("$P$", P, W); label("$Q$", Q, dir(270)); label("$R$", R, E); label("$S$", S, N); label("$a$", (A + P)/2, W); label("$a$", (B + Q)/2, dir(270)); label("$a$", (C + R)/2, E); label("$a$", (D + S)/2, N); [/asy][/asy]

With $2018$ points, a network composed of hexagons is formed as a sample the figure: [asy][asy] unitsize(1 cm); int i; path hex = dir(30)--(0,1)--dir(150)--dir(210)--(0,-1)--dir(330)--cycle; draw(hex); draw(shift((sqrt(3),0))*(hex)); draw(shift((2*sqrt(3),0))*(hex)); draw(shift((4*sqrt(3),0))*(hex)); draw(shift((5*sqrt(3),0))*(hex)); dot((3*sqrt(3) - 0.3,0)); dot((3*sqrt(3),0)); dot((3*sqrt(3) + 0.3,0)); dot(dir(150)); dot(dir(210)); for (i = 0; i <= 5; ++i) { if (i != 3) { dot((0,1) + i*(sqrt(3),0)); dot(dir(30) + i*(sqrt(3),0)); dot(dir(330) + i*(sqrt(3),0)); dot((0,-1) + i*(sqrt(3),0)); } } dot(dir(150) + 4*(sqrt(3),0)); dot(dir(210) + 4*(sqrt(3),0)); [/asy][/asy] A bee and a fly play the following game: initially the bee chooses one of the $2018$ dots and paints it red, then the fly chooses one of the $2017$ unpainted dots and paint it blue. Then the bee chooses an unpainted point and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If at the end of the game there is an equilateral triangle with red vertices, the bee wins, otherwise Otherwise the fly wins. Determine which of the two insects has a winning strategy.

Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

Consider the set $\Omega$ formed by the first twenty natural numbers, $\Omega = \{1, 2, . . . , 20\}$ . A nonempty subset $A$ of $\Omega$ is said to be sumfree if for all pair of elements$ x, y \in A$, the sum $(x + y)$ is not in $A$, ( $x$ can be equal to $y$). Prove that $\Omega$ has at least $2018$ sumfree subsets.

Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.