Hard inequality
by JK1603JK, May 1, 2025, 3:55 AM
Let 
and 
then prove 
and 
then prove 
Geometry Proof
by Jackson0423, Apr 30, 2025, 4:17 PM
In triangle 
, point 
on 
satisfies 
and 
.
Let
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
Then, the relation
holds.
Prove that is an isosceles triangle.
, point 
on 
satisfies 
and 
.Let
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.Then, the relation
holds.Prove that
is an isosceles triangle.Do not try to case bash lol
by ItzsleepyXD, Apr 30, 2025, 9:08 AM
Let 
be a positive integer such that 
.
Prove that
.
be a positive integer such that 
.Prove that
.Or statement function
by ItzsleepyXD, Apr 30, 2025, 9:07 AM
Find all 
such that 
for all real number 
and 
such that 
for all real number 
and 
Trivial fun Equilateral
by ItzsleepyXD, Apr 30, 2025, 9:05 AM
Let 
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .
Prove that
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .Prove that
BMO 2024 SL A5
by MuradSafarli, Apr 27, 2025, 12:44 PM
Let 
be the set of positive real numbers.
Find all non-negative real numbers
such that there exists a function 
with the property:
![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](//latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be the set of positive real numbers.Find all non-negative real numbers
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.Arbitrary point on BC and its relation with orthocenter
by falantrng, Apr 27, 2025, 11:47 AM
In an acute-angled triangle 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
Proposed by Theoklitos Parayiou, Cyprus
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.Proposed by Theoklitos Parayiou, Cyprus
This post has been edited 1 time. Last edited by falantrng, Apr 27, 2025, 4:38 PM
Something nice
by KhuongTrang, Nov 1, 2023, 12:56 PM
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM
N lines cutting each other in the plane
by M.J.Espinas, May 5, 2016, 7:00 PM
Let 
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
. We call an "Interior Point" if there are points 
on 
and 
on 
such that is between and . Prove that there are at least 
Interior points.(
)
note: by point here we mean the points which are intersection point of two of
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
. We call an "Interior Point" if there are points 
on 
and 
on 
such that is between and . Prove that there are at least 
Interior points.(
)note: by point here we mean the points which are intersection point of two of
.This post has been edited 1 time. Last edited by M.J.Espinas, May 5, 2016, 7:01 PM
APMO 2015 P1
by aditya21, Mar 30, 2015, 8:17 AM
Let be a triangle, and let 
be a point on side 
. A line through intersects side at and ray at . The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point distinct from point 
. The lines 
and 
intersect again at 
and respectively.
Prove that
Proposed by Warut Suksompong, Thailand
be a triangle, and let 
be a point on side 
. A line through intersects side at and ray at . The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point distinct from point 
. The lines 
and 
intersect again at 
and respectively.Prove that
Proposed by Warut Suksompong, Thailand
This post has been edited 1 time. Last edited by djmathman, Apr 5, 2015, 3:46 PM
March 2018
February 2018
January 2018