by Miquel-point, Apr 6, 2025, 7:57 PM 
so that the sum of its edges' lengths is minumum.by Miquel-point, Apr 6, 2025, 7:54 PM 
with 
equilateral. Let 
be the area of the triangle of sides 
, 
and 
. Show that 
where 
is the circumradius and 
is the volume of the tetrahedron.by Miquel-point, Apr 6, 2025, 7:50 PM 
where 
.
such that for every 
we have 
. is irrational then for every 
there exists 
such that![\[u<P(n)-\lfloor P(n)\rfloor <v.\]](http://latex.artofproblemsolving.com/9/4/a/94a8cb1f064e1dea37e8894fb4fd4e34c4c6c845.png)
Generalize the problem!by Miquel-point, Apr 6, 2025, 7:47 PM we consider a variable point 
which moves on line 
towards 
. Halfway there, it stops and starts moving in a straight line line towards 
. Halfway there, it stops and starts moving in a straight line towards 
, and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area 
.by Miquel-point, Apr 6, 2025, 6:15 PM 
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
by Mathdreams, Apr 6, 2025, 1:34 PM 
such that 
and 
for any real numbers 
and 
.by luutrongphuc, Apr 4, 2025, 5:35 AM be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .by a_507_bc, Apr 4, 2024, 4:26 PM be a triangle with incenter and -excenter 
, whose incircle touches 
at 
. The line 
meets at and is the midpoint of 
. Show that 
.by orl, Apr 20, 2006, 5:58 PM 
find, in terms of , the minimum value of 
for which there is a set of 
distinct positive integers that has sum greater than but every subset of size has sum at most 
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satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)