[/quote]
,
be an arbitrary triangle and 
is the circumcenter of 
.
lie on 
,
WRT 
is tangent to circumcircle of 
.
is tangent to circumcircle of triangle 
.
be an odd prime number. Let 
be the sequence defined as follows: 
,
,
is the smallest integer greater than 
such that in 
there are no arithmetic progressions of length 
.
with 
,
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
,
positive integers 
such that for every index 
,
,
is a multiple of 
.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
Y by Eugenis, rafayaashary1, joey8189681, wu2481632, JasperL, giacomorizzo, greenturtle3141, Wizard_32, fatant, Systematicworker, OlympusHero, ryanbear, Adventure10, Mango247, Aopamy, ihatemath123, and 13 other users
Hello, I was a participant at the 51st IMO in Kazakhstan this year.
I would like to use this thread to shed light on some misconceptions regarding the North Korean team and their disqualification.
Let me start off with a few quotes, from reliable sources.
1.
- A person on the coordinators' team.
2.
- A team leader.
3.
- Another team leader, information confirmed by an internet source.
4.
- U. Ri, a member of the North Korean team.
The first two quotes show the unjustness of North Korea's disqualification this year. Unlike North Korea's 1991 disqualification, the disqualification this year was not based on hard evidence. Instead, it was decided by a vote, which was probably fueled by suspicion. What kind of justice system bases its decision on a vote? It is widely accepted around the globe that people are innocent unless proven guilty. Is the IMO jury turning to frenzy and giving up reason? The IMO jury's decision this year is comparable to a modern witch trial. Too many factors added to the bias of the jury - North Korea's precedent in 1991, and its unbelievely high performance at the IMO this year, to name a few.
Now we suppose that North Korea did not cheat - what factors led to the participants using the same lemma? Look at quote 3 - the North Korean Math Olympians are trained separately for about 1~2years. Unlike America, or other countries like Japan or South Korea, they do not have access to local math programs, or valuable internet classes. The resources made available to these kids are basically the same. Although the North Korean students are creative intellectuals, they think the same way when it comes to math, for they have the exact same training. Also, we cannot completely rule out the possibility that the North Koreans had a very similar problem during one of their practice sessions.
As for me, the North Korean students came across as very bright kids, with a positive attitude toward life. (I would also like to say that Ri Yong Hyon is a party animal). I was next to the North Koreans at the precise moment they checked their results online (this was on Sunday night). At this time, it did not say disqualified next to North Korea; North Korea's page on the website was removed while the jury meeting was going on. One of the North Koreans kept on checking the page for North Korea, believing their test papers hadn't been fully graded yet. The next morning, one of them kept asking me whether I knew what was going on or not, and kept on saying they did not understand. If they had knowingly cheated, they wouldn't have reacted this way.
There are several measures that could be taken，none of them as radical as a disqualification. One thing that can be done is to reinstate the North Koreans' scores, yet make them unofficial. Another thing that can be done is to remove North Korea's scores for number 3. But I believe that the right thing to do is to restore North Korea's scores completely.
I would like to use this thread to shed light on some misconceptions regarding the North Korean team and their disqualification.
Let me start off with a few quotes, from reliable sources.
1.
Quote:
Nobody talked about flash cards, the only evidence of cheating was that almost all participants from North Korea started the problem 3 with the same lemma which is used in the official solution.
2.
Quote:
The North Koreans' disqualification was based on a vote of the jury
3.
Quote:
The North Korean education system consists of grades K-10; talented students in math are trained separately for the IMO.
4.
Quote:
I still don't what is going on here. Why are we disqualified?
The first two quotes show the unjustness of North Korea's disqualification this year. Unlike North Korea's 1991 disqualification, the disqualification this year was not based on hard evidence. Instead, it was decided by a vote, which was probably fueled by suspicion. What kind of justice system bases its decision on a vote? It is widely accepted around the globe that people are innocent unless proven guilty. Is the IMO jury turning to frenzy and giving up reason? The IMO jury's decision this year is comparable to a modern witch trial. Too many factors added to the bias of the jury - North Korea's precedent in 1991, and its unbelievely high performance at the IMO this year, to name a few.
Now we suppose that North Korea did not cheat - what factors led to the participants using the same lemma? Look at quote 3 - the North Korean Math Olympians are trained separately for about 1~2years. Unlike America, or other countries like Japan or South Korea, they do not have access to local math programs, or valuable internet classes. The resources made available to these kids are basically the same. Although the North Korean students are creative intellectuals, they think the same way when it comes to math, for they have the exact same training. Also, we cannot completely rule out the possibility that the North Koreans had a very similar problem during one of their practice sessions.
As for me, the North Korean students came across as very bright kids, with a positive attitude toward life. (I would also like to say that Ri Yong Hyon is a party animal). I was next to the North Koreans at the precise moment they checked their results online (this was on Sunday night). At this time, it did not say disqualified next to North Korea; North Korea's page on the website was removed while the jury meeting was going on. One of the North Koreans kept on checking the page for North Korea, believing their test papers hadn't been fully graded yet. The next morning, one of them kept asking me whether I knew what was going on or not, and kept on saying they did not understand. If they had knowingly cheated, they wouldn't have reacted this way.
There are several measures that could be taken，none of them as radical as a disqualification. One thing that can be done is to reinstate the North Koreans' scores, yet make them unofficial. Another thing that can be done is to remove North Korea's scores for number 3. But I believe that the right thing to do is to restore North Korea's scores completely.
![\[
p \mid a-b \iff p \mid g(a) - g(b),
\]](http://latex.artofproblemsolving.com/3/4/0/340922ec6b902c11209c06c0af4ef80285ce8180.png)
