A sharp one with 3 var (3)
by mihaig, May 27, 2025, 5:17 PM
Brilliant Problem
by M11100111001Y1R, May 27, 2025, 7:28 AM
Find all sequences
of natural numbers such that for every pair of natural numbers
and
, the following inequality holds:
![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](//latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)



![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
Cup of Combinatorics
by M11100111001Y1R, May 27, 2025, 7:24 AM
There are
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.









This post has been edited 1 time. Last edited by M11100111001Y1R, Tuesday at 7:26 AM
Bulgaria National Olympiad 1996
by Jjesus, Jun 10, 2020, 7:34 PM
Problem 4
by codyj, Jul 11, 2015, 6:30 AM
Triangle
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
.
Suppose that the lines
and
are different and intersect at the point
. Prove that
lies on the line
.
Proposed by Greece





























Suppose that the lines





Proposed by Greece
This post has been edited 3 times. Last edited by djmathman, Jun 14, 2018, 4:21 PM
Reason: italicized authors
Reason: italicized authors
Iran Inequality
by mathmatecS, Jun 11, 2015, 9:05 AM
When
satisfy
prove in equality.






This post has been edited 1 time. Last edited by mathmatecS, Jun 11, 2015, 9:31 AM
Can't be power of 2
by shobber, Mar 17, 2006, 3:26 PM
Show that for any positive integers
and
,
cannot be a power of
.




IMO96/2 [the lines AP, BD, CE meet at a point]
by Arne, Sep 30, 2003, 6:06 PM
Let
be a point inside a triangle
such that
![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](//latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
Let
,
be the incenters of triangles
,
, respectively. Show that the lines
,
,
meet at a point.


![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
Let







"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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