Found problems: 9
2017 International Olympic Revenge, 3
Let $ABC$ be a triangle, and let $P$ be a distinct point on the plane. Moreover, let $A'B'C'$ be a homothety of $ABC$ with ratio $2$ and center $P$, and let $O$ and $O'$ be the circumcenters of $ABC$ and $A'B'C'$, respectively. The circumcircles of $AB'C'$, $A'BC'$, and $A'B'C$ meet at points $X$, $Y$, and $Z$, different from $A'$, $B'$, and $C'$. In a similar way, the circumcircles of $A'BC$, $AB'C$, and $ABC'$ meet at $X'$, $Y'$, and $Z'$, different from $A$, $B$, $C$. Let $W$ and $W'$ be the circumcenters of $XYZ$ and $X'Y'Z'$, respectively. Prove that $OW$ is parallel to $O'W'$.
[i]Proposed by Mateus Thimóteo, Brazil.[/i]
2018 International Olympic Revenge, 1
Let $p$ be a prime number, and $X$ be the set of cubes modulo $p$, including $0$. Denote by $C_2(k)$ the number of ordered pairs $(x, y) \in X \times X$ such that $x + y \equiv k \pmod p$. Likewise, denote by $C_3(k)$ the number of ordered pairs $(x, y, z) \in X \times X \times X$ such that $x + y + z \equiv k \pmod p$.
Prove that there are integers $a, b$ such that for all $k$ not in $X$, we have
\[
C_3(k) = a\cdot C_2(k) + b.
\]
[i]Proposed by Murilo Corato, Brazil.[/i]
2017 International Olympic Revenge, 2
A polynomial is [i]good[/i] if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius $0.99$ on the complex plane that contains all the roots. Prove that there is no [i]good[/i] polynomial for a sufficient large degree.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.[/i]
2017 International Olympic Revenge, 1
Let $f(x)$ be the distance from $x$ to the nearest perfect square. For example, $f(\pi) = 4 - \pi$. Let $\alpha = \frac{3 + \sqrt{5}}{2}$ and let $m$ be an integer such that the sequence $a_n = f(m \; \alpha^n)$ is bounded. Prove that either $m=k^2$ or $m = 5k^2$ for some integer $k$.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil[/i].
2018 International Olympic Revenge, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]
2018 International Olympic Revenge, 3
When the IMO is over and students want to relax, they all do the same thing:
download movies from the internet. There is a positive number of rooms with internet
routers at the hotel, and each student wants to download a positive number of bits. The
load of a room is defined as the total number of bits to be downloaded from that room.
Nobody likes slow internet, and in particular each student has a displeasure equal to the
product of her number of bits and the load of her room. The misery of the group is
defined as the sum of the students’ displeasures.
Right after the contest, students gather in the hotel lobby to decide who goes to which
room. After much discussion they reach a balanced configuration: one for which no student
can decrease her displeasure by unilaterally moving to another room. The misery
of the group is computed to be $M_1$, and right when they seemed satisfied, Gugu arrived
with a serendipitous smile and proposed another configuration that achieved misery $M_2$.
What is the maximum value of $M_1/M_2$ taken over all inputs to this problem?
[i]Proposed by Victor Reis (proglote), Brazil.[/i]
2017 International Olympic Revenge, 4
Let $n>1$ be a positive integer. Ana and Bob play a game with other $n$ people. The group of $n$ people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.
Before Bob distribute their hats, Ana gives $n$ people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white".
Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary.
Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee?
[i]Proposed by China.[/i]
BIMO 2020, 3
Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively.
Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles
$(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$.
[i]Proposed by Ivan Chai, Malaysia.[/i]
2018 International Olympic Revenge, 2
Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively.
Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles
$(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$.
[i]Proposed by Ivan Chai, Malaysia.[/i]