[/quote]
,
![$1+[\sqrt{2n}]~$](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 
x
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
,
,
respectively. Let 
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
let 
denote the intersection of diagonals 
and 
,
denote the intersection of diagonals 
and 
.
and 
,
bisects angle 
.
are 
,
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 
,
has a closed walk of length at most 
passing through each edge of 
at least once.
Y by Irreplaceable
Let 
be a positive integer. Two players 
and 
play a game in which they take turns choosing positive integers 
. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any number chosen by any player on any turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
be a positive integer. Two players 
and 
play a game in which they take turns choosing positive integers 
. The rules of the game are:(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any number chosen by any player on any turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player
takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
or 
or 
(sorry for game loss).
