Hard Inequality
by danilorj, May 20, 2025, 5:17 AM
Let 
with 
. Prove that:
![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](//latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.
with 
. Prove that:![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.This post has been edited 1 time. Last edited by danilorj, Today at 5:18 AM
Reason: ..
Reason: ..
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
and 
lie on opposite sides of 
,
and 
lie on the opposite siedes of 
.
.
Where 
Computing functions
by BBNoDollar, May 18, 2025, 5:25 PM
Let 
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](//latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
.
easy geo
by ErTeeEs06, Apr 26, 2025, 11:13 AM
Let 
be a triangle with incentre 
and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
be a triangle with incentre 
and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.This post has been edited 1 time. Last edited by ErTeeEs06, Apr 26, 2025, 11:15 AM
Reason: latex
Reason: latex
Where are the Circles?
by luminescent, Apr 9, 2022, 10:00 PM
Let be an acute-angled triangle in which 
and 
. Let point 
lie on segment 
and point 
lie on segment 
such that 
, 
and 
. Let 
be the circumcenter of triangle 
, 
the orthocenter of triangle , and 
the point of intersection of the lines 
and 
. Prove that , , and are collinear.
be an acute-angled triangle in which 
and 
. Let point 
lie on segment 
and point 
lie on segment 
such that 
, 
and 
. Let 
be the circumcenter of triangle 
, 
the orthocenter of triangle , and 
the point of intersection of the lines 
and 
. Prove that , , and are collinear.This post has been edited 1 time. Last edited by luminescent, Apr 9, 2022, 10:07 PM
Reason: change source format to match other egmo problems
Reason: change source format to match other egmo problems
Number of Solutions is 2
by Miku3D, Jun 9, 2021, 6:29 AM
Prove that for each real number 
, there are exactly two or three positive real numbers 
satisfying the equation 
.
, there are exactly two or three positive real numbers 
satisfying the equation 
.IMO 2016 Shortlist, N6
by dangerousliri, Jul 19, 2017, 4:31 PM
Denote by 
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
, the integer 
is nonzero and divides 
.
Proposed by Dorlir Ahmeti, Albania
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
, the integer 
is nonzero and divides 
.Proposed by Dorlir Ahmeti, Albania
March 2009
February 2009
January 2009