[/quote]
,
.
amount of people stand on coordinates 
where 
.
.
got 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
litres of water.
![$1+[\sqrt{2n}]~$](http://latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.
, ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding 
let 
denote the intersection of diagonals 
and 
,
denote the intersection of diagonals 
.
and 
,
bisects angle 
.
x
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
be the projection of 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
t
,
is 
and 
,
denote the circumradius. Prove that if 
,
.
-
dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the 
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
,
Y by
Let 
be a point on the side 
of the triangle 
such that 
, and let the points 
on the side 
and 
on the extension 
beyond the point 
be such that 
. The lines 
and 
intersect at the point 
. Prove that is the bisector of the angle 
.
be a point on the side 
of the triangle 
such that 
, and let the points 
on the side 
and 
on the extension 
beyond the point 
be such that 
. The lines 
and 
intersect at the point 
. Prove that is the bisector of the angle 
.This post has been edited 1 time. Last edited by parmenides51, Jun 27, 2022, 9:30 AM
