G
Topic
First Poster
Last Poster
Fractional Inequality
sqing 33
N
by Learning11
Source: Chinese Girls Mathematical Olympiad 2012, Problem 1
Let 
be non-negative real numbers. Prove that
be non-negative real numbers. Prove that
33 replies
Geometry angle chasing olympiads
Foxellar 1
N
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 
.
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
by Learning11
Source: Iran 1998
When 
satisfy 
prove in equality.
satisfy 
prove in equality.
17 replies
Problem 4
codyj 87
N
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
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
87 replies
IMO96/2 [the lines AP, BD, CE meet at a point]
Arne 47
N
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.
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)
Let
, 
be the incenters of triangles 
, 
, respectively. Show that the lines 
, 
, 
meet at a point.
47 replies
satisfying
Prove
Cup of Combinatorics
M11100111001Y1R 1
N
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.
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
by reni_wee
Find all prime numbers 
for which 
divides 
.
for which 
divides 
.
7 replies
Can't be power of 2
shobber 31
N
by LeYohan
Source: APMO 1998
Show that for any positive integers 
and 
, 
cannot be a power of 
.
and 
, 
cannot be a power of 
.
31 replies
Brilliant Problem
M11100111001Y1R 4
N
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:
![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](//latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
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
