AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
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. will not have more than 
litres of water.
+1 w such that ![$1+[\sqrt{2n}]~$](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 
, and let 
denote the intersection of diagonals 
and 
. Prove that if 
and 
, then 
bisects angle 
.x table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a 
x square which contains an odd number of unit squares white colored.
be a scalene triangle with circumcenter 
, incenter 
, and incircle 
. Let touch the sides 
, 
, and 
at points 
, 
, and 
respectively. Let 
be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
th power.
are and 
, respectively. The excenter and exradius relative to vertex 
is 
and 
, respectively. Let 
denote the circumradius. Prove that if 
, then 
.
-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 
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 dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
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 
, respectively.
is a positive integer, then 
can be written as a finite sum of reciprocals of different triangular numbers.
for some positive integer 
.]
is a positive integer, then can be written as a finite sum of reciprocals of different triangular numbers. for some positive integer .]
.
as desired. 