Olympiad problems

2021 EGMO, 1

The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?

2021 EGMO, 2

Tags: EGMO , functional equation , EGMO 2021 , Hi

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation \[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$. Here, $\mathbb{Q}$ denotes the set of rational numbers.

2021 EGMO, 4

Tags: EGMO 2021 , geometry , reflection , EGMO , Triangle

Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.

2021 EGMO, 5

Tags: EGMO 2021 , geometry , combinatorial geometry , EGMO , Triangles

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that [list] [*] no three points in $P$ lie on a line and [*] no two points in $P$ lie on a line through the origin. [/list] A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.

2021 EGMO, 3

Tags: EGMO 2021 , geometry , Triangle , EGMO

Let $ABC$ 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 $ABC$ through $B$ and $C$ respectively. Let $M$ and $N$ be the points on the segments $EC$ and $FB$ respectively such that $\angle EMA = \angle BCA$ and $\angle ANF = \angle ABC$. Prove that the points $E, F, N, M$ lie on a circle.

2021 EGMO, 6

Tags: EGMO 2021 , number theory , EGMO , asymptotics , algebra , equation , counting

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? [i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]