Inequality in triangle
by Nguyenhuyen_AG, May 31, 2025, 6:17 AM
Let 
be the lengths of the sides of a triangle. Prove that
![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](//latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
Tough inequality
by TUAN2k8, May 28, 2025, 2:03 AM
Let 
be an even integer and let 
be real numbers satisfying 
.
Prove that
be an even integer and let 
be real numbers satisfying 
.Prove that
This post has been edited 1 time. Last edited by TUAN2k8, Today at 1:23 AM
Reason: .
Reason: .
Symmetric points part 2
by CyclicISLscelesTrapezoid, Jun 27, 2022, 4:00 PM
Let 
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle 
. The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .
Define
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.
Hongzhou Lin
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle 
. The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .Define
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.Hongzhou Lin
This post has been edited 1 time. Last edited by CyclicISLscelesTrapezoid, Jun 27, 2022, 4:08 PM
.
Converse of a classic orthocenter problem
by spartacle, Dec 14, 2020, 5:00 PM
Let 
, 
, 
, 
be four points such that no three are collinear and is not the orthocenter of . Let 
, 
, 
be the orthocenters of 
, 
, 
, respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.
Andrew Gu
, 
, 
, 
be four points such that no three are collinear and is not the orthocenter of . Let 
, 
, 
be the orthocenters of 
, 
, 
, respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.Andrew Gu
This post has been edited 2 times. Last edited by v_Enhance, Mar 1, 2021, 5:21 PM
such that 
Problem 1
by randomusername, Jul 10, 2015, 8:12 AM
We say that a finite set 
of points in the plane is balanced if, for any two different points and in , there is a point in such that 
. We say that is centre-free if for any three different points , and in , there is no points in such that 
.
(a) Show that for all integers
, there exists a balanced set consisting of 
points.
(b) Determine all integers for which there exists a balanced centre-free set consisting of points.
Proposed by Netherlands
of points in the plane is balanced if, for any two different points and in , there is a point in such that 
. We say that is centre-free if for any three different points , and in , there is no points in such that 
.(a) Show that for all integers
, there exists a balanced set consisting of 
points.(b) Determine all integers
for which there exists a balanced centre-free set consisting of points.Proposed by Netherlands
This post has been edited 3 times. Last edited by v_Enhance, Jul 26, 2015, 2:46 PM
Reason: Missing $n$ in part (b)
Reason: Missing $n$ in part (b)
x is rational implies y is rational
by pohoatza, Jun 28, 2007, 7:24 PM
For 
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.
Proposed by J.P. Grossman, Canada
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.Proposed by J.P. Grossman, Canada
Multiplicative function
by Tales, Mar 23, 2005, 2:50 AM
The function 
from the set 
of positive integers into itself is defined by the equality ![\[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]](//latex.artofproblemsolving.com/2/a/5/2a5a20395450d37d0352ab75e9e4875ab7803805.png)
a) Prove that
for every two relatively prime 
.
b) Prove that for each
the equation 
has a solution.
c) Find all
such that the equation has a unique solution.
from the set 
of positive integers into itself is defined by the equality ![\[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]](http://latex.artofproblemsolving.com/2/a/5/2a5a20395450d37d0352ab75e9e4875ab7803805.png)
a) Prove that
for every two relatively prime 
.b) Prove that for each
the equation 
has a solution.c) Find all
such that the equation has a unique solution.This post has been edited 2 times. Last edited by djmathman, Aug 1, 2015, 2:54 AM
Reason: formatting
Reason: formatting
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