Found problems: 4
2020 GQMO, 7
Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$
such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
[i]Massimiliano Foschi, Italy[/i]
2020 GQMO, 8
Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.
[i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]
2020 GQMO, 5
Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$
[i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]
2020 GQMO, 4
Prove that, for all sufficiently large integers $n$, there exists $n$ numbers $a_1, a_2, \dots, a_n$ satisfying the following three conditions:
[list]
[*] Each number $a_i$ is equal to either $-1, 0$ or $1$.
[*] At least $\frac{2n}{5}$ of the numbers $a_1, a_2, \dots, a_n$ are non-zero.
[*] The sum $\frac{a_1}{1} + \frac{a_2}{2} + \dots + \frac{a_n}{n}$ is $0$.
[/list]
$\textit{Note: Results with 2/5 replaced by a constant } c \textit{ will be awarded points depending on the value of } c$
[i]Proposed by Navneel Singhal, India; Kyle Hess, USA; and Vincent Jugé, France[/i]