Inequality from China
by sqing, Apr 3, 2025, 1:11 PM
Let 
Prove that 
Where 
Prove that 
Where 
This post has been edited 1 time. Last edited by sqing, Yesterday at 1:12 PM
pretty well known
by dotscom26, Apr 3, 2025, 2:03 AM
Let 
be a scalene triangle such that 
is its incircle. 
is tangent to at 
. A point 
(
) is located on 
.
Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.
Show that the common tangent to and is also tangent to .
be a scalene triangle such that 
is its incircle. 
is tangent to at 
. A point 
(
) is located on 
.Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.Show that the common tangent to
and is also tangent to .
such that 
Proving ZA=ZB
by nAalniaOMliO, Mar 28, 2025, 8:36 PM
Point 
is the foot of the altitude from 
of triangle 
. On the lines and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
Prove that
.
Vadzim Kamianetski
is the foot of the altitude from 
of triangle 
. On the lines and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.Prove that
.Vadzim Kamianetski
This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 6:45 PM
Unsolved NT, 3rd time posting
by GreekIdiot, Mar 26, 2025, 11:40 AM
Solve 
where 
where 
This post has been edited 2 times. Last edited by GreekIdiot, Mar 26, 2025, 11:41 AM
NMO (Nepal) Problem 4
by khan.academy, Mar 17, 2024, 2:24 PM
Find all integer/s 
such that 
is a prime or a perfect square of an integer.
Proposed by Prajit Adhikari, Nepal
such that 
is a prime or a perfect square of an integer.Proposed by Prajit Adhikari, Nepal
This post has been edited 1 time. Last edited by khan.academy, Mar 17, 2024, 2:39 PM
2019 Nepal National Mathematics Olympiad
by Piinfinity, Oct 13, 2020, 5:56 AM
Problem 31
Let
be a function such that
for all real numbers
. Determine 
.
Let
be a function such that
for all real numbers
. Determine 
.Wot n' Minimization
by y-is-the-best-_, Sep 23, 2020, 12:00 AM
Let 
be a positive integer and let 
be a strictly increasing sequence of positive real numbers with sum equal to 2. Let be a subset of 
such that the value of
![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](//latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of positive real numbers 
with sum equal to 2 such that
![\[
\sum_{i \in X} b_{i}=1.
\]](//latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a positive integer and let 
be a strictly increasing sequence of positive real numbers with sum equal to 2. Let be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of positive real numbers 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
Kosovo Mathematical Olympiad 2016 TST , Problem 1
by dangerousliri, Jan 9, 2017, 3:38 PM
Solve equation in real numbers
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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