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Fractional Inequality
sqing 33
N
an hour ago
by Learning11
Source: Chinese Girls Mathematical Olympiad 2012, Problem 1
Let
be non-negative real numbers. Prove that



33 replies
Geometry angle chasing olympiads
Foxellar 1
N
an hour ago
by Ianis
Let
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
.








1 reply
Iran Inequality
mathmatecS 17
N
an hour ago
by Learning11
Source: Iran 1998
When
satisfy
prove in equality.





17 replies
Problem 4
codyj 87
N
2 hours ago
by ezpotd
Source: IMO 2015 #4
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
87 replies

IMO96/2 [the lines AP, BD, CE meet at a point]
Arne 47
N
3 hours ago
by Bridgeon
Source: IMO 1996 problem 2, IMO Shortlist 1996, G2
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







47 replies
A sharp one with 3 var (3)
mihaig 4
N
3 hours ago
by aaravdodhia
Source: Own
Let
satisfying
Prove



4 replies
Cup of Combinatorics
M11100111001Y1R 1
N
4 hours ago
by Davdav1232
Source: Iran TST 2025 Test 4 Problem 2
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.









1 reply
Bulgaria National Olympiad 1996
Jjesus 7
N
4 hours ago
by reni_wee
Find all prime numbers
for which
divides
.



7 replies
Can't be power of 2
shobber 31
N
4 hours ago
by LeYohan
Source: APMO 1998
Show that for any positive integers
and
,
cannot be a power of
.




31 replies
Brilliant Problem
M11100111001Y1R 4
N
4 hours ago
by IAmTheHazard
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences
of natural numbers such that for every pair of natural numbers
and
, the following inequality holds:



![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
4 replies
