EGMO 2021

According to Anna, the number 2021 is fantabulous. She states that if any element of the set $\{m, 2 m+1,3 m\}$ is fantabulousfor a positive integer $m$, then they are all fantabulous. Is the number 2021 ${ }^{2021}$ fantabulous? (Australia, Angelo Di Pasquale)

Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation

$$f(x f(x)+y)=f(y)+x^{2}$$

holds for all rational numbers $x$ and $y$. Here, $\mathbb{Q}$ denotes the set of rational numbers. (Slovakia, Patrik Bak)

Let $A B C$ be a triangle with an obtuse angle at $A$. Let $E$ and $F$ be the intersections of the external bisector of angle $A$ with the altitudes of $A B C$ through $B$ and $C$ respectively. Let $M$ and $N$ be the points on the segments $E C$ and $F B$ respectively such that $\angle E M A=\angle B C A$ and $\angle A N F=\angle A B C$. Prove that the points $E, F, N, M$ lie on a circle.

Let $A B C$ be a triangle with incentre Iand let $D$ be an arbitrary point on the side $B C$. Let the line through $D$ perpendicular to $B I$ intersect $C I$ at $E$. Let the line through $D$ perpendicular to $C I$ intersect $B I$ at $F$. Prove that the reflection of $A$ in the line $E F$ lies on the line $B C$. (Australia, Sampson Wong)

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that (i) no three points in $P$ lie on a line and (ii) no two points in $P$ lie on a line through the origin.

A triangle with vertices in $P$ is fat, if $O$ is strictly inside the triangle. Find the maximum number of fat triangles. (Austria, Veronika Schreitter)

Does there exist a nonnegative integer $a$ for which the equation

$$\left\lfloor\frac{m}{1}\right\rfloor+\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{m}{3}\right\rfloor+\cdots+\left\lfloor\frac{m}{m}\right\rfloor=n^{2}+a$$

has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers? (The expression $\lfloor x\rfloor$ denotesthe integer part(or floor) of the real numberx. Thus $\lfloor\sqrt{2}\rfloor=1,\lfloor\pi\rfloor=\left\lfloor\frac{22}{7}\right\rfloor=$ $3,\lfloor 42\rfloor=42$ and $\lfloor 0\rfloor=0)$