by Foxellar, May 28, 2025, 10:32 PM 
be a triangle such that 
. Points 
lie on segments 
, respectively, such that lines 
and 
are the angle bisectors of triangle 
. Find the measure of angle 
.by M11100111001Y1R, May 27, 2025, 7:28 AM 
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
by M11100111001Y1R, May 27, 2025, 7:24 AM 
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
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.by Jjesus, Jun 10, 2020, 7:34 PM 
for which 
divides 
.by codyj, Jul 11, 2015, 6:30 AM 
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment 
at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
. Prove that lies on the line 
.by mathmatecS, Jun 11, 2015, 9:05 AM 
satisfy 
prove in equality.
by shobber, Mar 17, 2006, 3:26 PM 
and 
, 
cannot be a power of 
.by Arne, Sep 30, 2003, 6:06 PM 
be a point inside a triangle 
such that![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
, 
be the incenters of triangles 
, 
, respectively. Show that the lines 
, 
, 
meet at a point."Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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satisfying
Prove
be non-negative real numbers. Prove that
