satisfying
Prove
be an acute traingle with 
.
is a circumcenter of 
and 
are equal.
.
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 
Y by Adventure10
On a circle with diameter 
, let 
be an arbitrary point distinct from 
and 
. Points 
are the midpoints of chords 
, and points 
are the midpoints of smaller arcs restricted by these chords. Lines 
and 
meet at point 
, and lines 
and 
meet at point 
. Prove that lines 
and 
concur.
, let 
be an arbitrary point distinct from 
and 
. Points 
are the midpoints of chords 
, and points 
are the midpoints of smaller arcs restricted by these chords. Lines 
and 
meet at point 
, and lines 
and 
meet at point 
. Prove that lines 
and 
concur.
as the reference triangle. We have,
Since 
is a arc midpoint, it follows that 
Plugging this into the circumcircle equation gives,
In a similar fashion we have
Trivially,
Also,
Therefore, the equation for line 
The equation for line 
is
The equation for line 
is
By the concurrency lemma, it suffices to show that
However, the determinant in question turns out to be
by the Pythagorean Theorem.
and 
.
