Inspired by hlminh
by sqing, Apr 22, 2025, 8:09 AM
Let 
be real numbers such that 
Prove that
Where 
Where 
be real numbers such that 
Prove that
Where 
Where 
Combo with cyclic sums
by oVlad, Apr 21, 2025, 1:41 PM
In 
of the vertices of the regular polygon 
we write the number 
and in the remaining ones we write the number 
Let 
be the number written on the vertex 
A vertex is good if ![\[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]](//latex.artofproblemsolving.com/b/f/4/bf4cf984af7f310c1eb90b6b37126bfca9181305.png)
for any integers 
and 
such that 
Note that the indices are taken modulo 
Determine the greatest possible value of such that, regardless of numbering, there always exists a good vertex.
of the vertices of the regular polygon 
we write the number 
and in the remaining ones we write the number 
Let 
be the number written on the vertex 
A vertex is good if ![\[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]](http://latex.artofproblemsolving.com/b/f/4/bf4cf984af7f310c1eb90b6b37126bfca9181305.png)
for any integers 
and 
such that 
Note that the indices are taken modulo 
Determine the greatest possible value of such that, regardless of numbering, there always exists a good vertex.Number theory
by XAN4, Apr 20, 2025, 9:44 AM
Prove that there exists infinitely many positive integers 
such that 
and 
.
such that 
and 
.Stronger inequality than an old result
by KhuongTrang, Aug 1, 2024, 3:13 PM
Problem. Find the best constant 
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](//latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
This post has been edited 2 times. Last edited by KhuongTrang, Aug 2, 2024, 6:43 AM
R+ FE with arbitrary constant
by CyclicISLscelesTrapezoid, Jul 5, 2023, 7:22 PM
Let 
be a given positive real and 
be the set of all positive reals. Find all functions 
such that ![\[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]](//latex.artofproblemsolving.com/b/0/6/b069e7b7ec1e277f4a4ce85a99434fdb54eb02f3.png)
be a given positive real and 
be the set of all positive reals. Find all functions 
such that ![\[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]](http://latex.artofproblemsolving.com/b/0/6/b069e7b7ec1e277f4a4ce85a99434fdb54eb02f3.png)
This post has been edited 2 times. Last edited by CyclicISLscelesTrapezoid, Jul 5, 2023, 7:24 PM
JBMO TST Bosnia and Herzegovina 2022 P3
by Motion, May 21, 2022, 9:37 PM
Let 
be an acute triangle. Tangents on the circumscribed circle of triangle at points 
and 
intersect at point 
. Let 
and 
be a foot of the altitudes from onto 
and 
and let 
be the midpoint of 
. Prove:
A) Prove that is the orthocenter of the triangle 
.
B) Prove that
cuts 
in half.
be an acute triangle. Tangents on the circumscribed circle of triangle at points 
and 
intersect at point 
. Let 
and 
be a foot of the altitudes from onto 
and 
and let 
be the midpoint of 
. Prove:A) Prove that
is the orthocenter of the triangle 
.B) Prove that
cuts 
in half.Inequality with n-gon sides
by mihaig, Feb 25, 2022, 6:46 AM
If 
are are the lengths of the sides of a 
gon such that
then
![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](//latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
When do we have equality?
(V. Cîrtoaje and L. Giugiuc, 2021)
are are the lengths of the sides of a 
gon such that
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
When do we have equality?
(V. Cîrtoaje and L. Giugiuc, 2021)
Incircle of a triangle is tangent to (ABC)
by amar_04, Mar 2, 2021, 3:36 AM
Let be a scalene triangle, 
be the median through 
, and 
be the incircle. Let touch at point and segment 
meet for the second time at point 
. Let 
be the triangle formed by lines and and the tangent to at . Prove that the incircle of triangle is tangent to the circumcircle of triangle .
be a scalene triangle, 
be the median through 
, and 
be the incircle. Let touch at point and segment 
meet for the second time at point 
. Let 
be the triangle formed by lines and and the tangent to at . Prove that the incircle of triangle is tangent to the circumcircle of triangle .
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