Inequality
by SunnyEvan, Jul 26, 2025, 9:00 AM
Let
Prove that:

When does the equality holds ?



This post has been edited 1 time. Last edited by SunnyEvan, 3 hours ago
2-var inequality
by sqing, Jul 26, 2025, 3:12 AM
Let
Prove that
Let
Prove that
Let
Prove that







This post has been edited 3 times. Last edited by sqing, Today at 3:33 AM
Peru IMO TST 2023
by diegoca1, Jul 25, 2025, 7:22 PM
Let
be non-negative real numbers such that
. Prove the inequality
and determine when equality holds.


![\[
6xyz \leq x(1 - x) + y(1 - y) + z(1 - z),
\]](http://latex.artofproblemsolving.com/f/5/b/f5b208ea6439994cb40f6b7f0806c216d9868e86.png)
D1053 : Set of Dirichlet
by Dattier, Jul 22, 2025, 1:23 PM
We say a set
have the Dirichlet propriety, if
,
.
Let
with
subset of
and have the Dirichlet propriety.
1) Is it true that
?
2) Is it true that
?



Let



1) Is it true that

2) Is it true that

Amazing rook and chessboard question
by egxa, Dec 17, 2024, 8:09 AM
Let
be positive integers. On an
chessboard, some unit squares are occupied by rooks such that each rook attacked by odd number of other rooks. Determine the maximum number of rooks that can be placed on the chessboard.


Symmetric inequality
by nexu, Feb 12, 2023, 6:54 AM
Iran geometry
by Dadgarnia, Mar 11, 2020, 2:07 PM
Given a triangle
with circumcircle
. Points
and
are the foot of angle bisectors of
and
,
is incenter and
is the intersection of
and
. Suppose that
be the midpoint of arc
. Circle
intersects the
-median and circumcircle of
for the second time at
and
. Let
be the reflection of
across
and
be the second intersection of circumcircle of
and
. Prove that quadrilateral
is cyclic.
Proposed by Alireza Dadgarnia and Amir Parsa Hosseini
























Proposed by Alireza Dadgarnia and Amir Parsa Hosseini
This post has been edited 1 time. Last edited by Dadgarnia, Mar 12, 2020, 10:38 AM
A sequence must be bounded
by math90, Jul 10, 2018, 11:08 AM
A sequence of real numbers
satisfies the relation
Prove that the sequence is bounded, i.e., there is a constant
such that
for all positive integers
.





This post has been edited 2 times. Last edited by math90, Jul 11, 2018, 2:47 PM
Reason: ISL=IMO Shortlist
Reason: ISL=IMO Shortlist
Tangent intersect intersect tangent intersect
by cjquines0, May 26, 2017, 11:12 AM
Let two circles
and
intersect in points
and
. The tangent to
at
intersects
in
and the line
intersects
for the second time in
(suppose that
is outside
). The tangent to
from
intersects
and
in
and
, respectively. (The points
and
lie on different sides of the line
.) Show that
is the bisector of
.
Proposed by Iman Maghsoudi
























Proposed by Iman Maghsoudi
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