OLYMPIC-REVENGE 2023

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that $$f^n(x)+f^n(y)=x+y$$

Find all triples ( $a$ , $b$ , $n$ ) of positive integers such that $$a^3=b^2+2^n$$

Define a $\emph{big circle}$ in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose $(AB)$ as the big circle that passes through $A$ and $B$ . Also, let a $\emph{Spheric Triangle}$ be $3$ connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also $\angle XYZ$ the angle between $(XY)$ and $(YZ)$ . Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S. Let $\Delta ABC$ be a spheric triangle with all its angles $<90^{\circ}$ such that there is a circle $\omega$ tangent to $(BC)$ , $(CA)$ , $(AB)$ in $D,E,F$ . Show that there is $P\in S$ with $\angle PAB=\angle DAC$ , $\angle PCA=\angle FCB$ , $\angle PBA=\angle EBC$ .

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$ . Is it possible for Gugu to cover all points of $S$ with his spheres?