cups labeled 
,
-
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
for which 
divides 
.
and 
,
cannot be a power of 
.
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
Y by
Let 
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .This post has been edited 1 time. Last edited by Assassino9931, Apr 9, 2025, 10:36 PM
at 
and let 
.
is the pole of 
and 
is the pole 
,
is the pole of 
and 
are collinear. Let 
and 
;
is harmonic, and since 
touch 
.
by projecting 
through 
,
.
such that the line 
passes through 
concur on 
,
,
concur, as desired.
