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be real numbers such that 
.
ISI UGB 2025 P4
SomeonecoolLovesMaths 9
N
by nyacide
Source: ISI UGB 2025 P4
Let 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
(Hint :
)
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.(Hint :
)
9 replies
Self-evident inequality trick
Lukaluce 9
N
by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
9 replies
Prove n is square-free given divisibility condition
CatalanThinker 1
N
by CatalanThinker
Source: 1995 Indian Mathematical Olympiad
Let 
be a positive integer such that divides the sum
![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](//latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that is square-free.
be a positive integer such that divides the sum![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that is square-free.
1 reply
.
Where 
What is thiss
EeEeRUT 5
N
by MathLuis
Source: Thailand MO 2025 P6
Find all function 
,such that the inequality 
holds for all positive reals 
and for every positive real 
, there exist positive reals 
, such that the equality holds.
,such that the inequality 
holds for all positive reals 
and for every positive real 
, there exist positive reals 
, such that the equality holds.
5 replies
Thailand geometry
EeEeRUT 4
N
by MathLuis
Source: Thailand MO 2025 P7
Let 
be a triangle with 
. The tangent to the circumcircle of 
at 
intersects 
at 
. The angle bisector of 
intersect at 
. Suppose that the perpendicular bisector of 
intersect 
at 
, respectively. Show that 
be a triangle with 
. The tangent to the circumcircle of 
at 
intersects 
at 
. The angle bisector of 
intersect at 
. Suppose that the perpendicular bisector of 
intersect 
at 
, respectively. Show that 
4 replies
JBMO Shortlist 2021 G2
Lukaluce 10
N
by Adventure1000
Source: JBMO Shortlist 2021
Let 
be an interior point of the isosceles triangle with 
. If
prove that 
.
Proposed by Mehmet Akif Yıldız, Turkey
be an interior point of the isosceles triangle with 
. If
prove that 
.Proposed by Mehmet Akif Yıldız, Turkey
10 replies
Thailand MO 2025 P3
Kaimiaku 5
N
by MathLuis
Let 
be positive real numbers such that 
. Prove that 
be positive real numbers such that 
. Prove that 
5 replies
Simple inequality
sqing 7
N
by sqing
Source: 2016 China Sichuan High School Mathematics Competition ，Q14
Let 
are positive real numbers .Show that
are positive real numbers .Show that
7 replies
