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First Poster
Last Poster
Easy IMO 2023 NT
799786 137
N
an hour ago
by InftyByond
Source: IMO 2023 P1
Determine all composite integers
that satisfy the following property: if
,
,
,
are all the positive divisors of
with
, then
divides
for every
.










137 replies
JBMO TST Bosnia and Herzegovina 2025 P3
Rotten_ 7
N
an hour ago
by sqing
Let a, b, c be real numbers such that
Find the minimum value of the expression
![\[
a + b + c = 0 \quad \text{and} \quad abc = -16.
\]](http://latex.artofproblemsolving.com/6/b/1/6b1532d4de660e7ab4300ee1f7ec1742cbde9dc2.png)
![\[
W = \frac{a^2 + b^2} {c} + \frac{b^2 + c^2} {a} + \frac{c^2 + a^2} {b}.
\]](http://latex.artofproblemsolving.com/2/2/3/223e5abc7ae4f5a92dde90a74b0988adb18bdf4e.png)
7 replies
Prove two circles intersect on line BC
62861 76
N
an hour ago
by KevinYang2.71
Source: USA Winter TST for IMO 2019, Problem 1 and TST for EGMO 2019, Problem 2, by Merlijn Staps
Let
be a triangle and let
and
denote the midpoints of
and
, respectively. Let
be a point such that
is tangent to the circumcircle of triangle
. Denote by
the circle through
and
tangent to
, and by
the circle through
and
tangent to
. Show that
and
intersect on line
.
Merlijn Staps



















Merlijn Staps
76 replies

Interesting inequality
sqing 5
N
an hour ago
by sqing
Source: Own
Let
and
Prove that
Where 







5 replies
Some Integers written in a black board
Kunihiko_Chikaya 9
N
2 hours ago
by Namura
Source: 2009 Japan Mathematical Olympiad Finals, Problem 2
Let
be postive integer. Some integers are written in a black board and those satisfy the following conditions.
1. Any numbers written are integers which are from 1 to
.
2. More than one integer which is from 1 to
is written.
3. The sum of numbers written is even.
If we mark
to some numbers written and mark
to all remaining numbers, then prove that we can set the sum of numbers marked
are equal to that of numbers marked
.

1. Any numbers written are integers which are from 1 to

2. More than one integer which is from 1 to

3. The sum of numbers written is even.
If we mark




9 replies

A tournament
Omid Hatami 22
N
2 hours ago
by de-Kirschbaum
Source: Iran TST 2008
Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.
22 replies
Symmetric inequality with two directions
Tintarn 23
N
3 hours ago
by Sakura-junlin
Source: Germany 2017, Problem 5
Prove that for all non-negative numbers
satisfying
, one has


![\[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]](http://latex.artofproblemsolving.com/d/9/c/d9cb40fd1427d3ce31af7525728b9d9a5031e2c6.png)
23 replies
Guessing Point is Hard
MarkBcc168 33
N
3 hours ago
by HamstPan38825
Source: IMO Shortlist 2023 G5
Let
be an acute-angled triangle with circumcircle
and circumcentre
. Points
and
lie on
such that
and
. Let
meet
at
, and
meet
at
.
Prove that the circumcircles of triangles
and
have an intersection lie on line
.
Ivan Chan Kai Chin, Malaysia














Prove that the circumcircles of triangles



Ivan Chan Kai Chin, Malaysia
33 replies
The only 2D geometry on the test
v_Enhance 17
N
4 hours ago
by KnowingAnt
Source: USA TSTST 2025/9
Let acute triangle
have orthocenter
. Let
,
,
, and
be collinear points which lie on lines
,
,
, and
, respectively. Let
and
be the circumcircles of triangles
and
, respectively. Prove that the radical axis of
and
intersects the line through their centers on the nine-point circle of triangle
.
Ruben Carpenter

















Ruben Carpenter
17 replies
