be nonnegative real numbers such that 
Find the minimum value of 
be positive reals such that 
Find the maximum value of ![$$x_1+\sqrt{x_1x_2}+\sqrt[3]{x_1x_2x_3}+\sqrt[4]{x_1x_2x_3x_4}+\sqrt[5]{x_1x_2x_3x_4x_5}.$$](http://latex.artofproblemsolving.com/2/5/f/25f59ee5503bb04b51590698e5fe2b8352828ddc.png)
.
be a subset of 
such that no sum of three distinct elements of 
.
.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
Y by Adventure10, Mango247
Let 
be a triangle such that 
. Let 
be a point about 
. Let 
be the feet of the perpendiculars from to 
and 
respectively. Show that the value of the sum 
it does not depend on the position of the chosen point .
be a triangle such that 
. Let 
be a point about 
. Let 
be the feet of the perpendiculars from to 
and 
respectively. Show that the value of the sum 
it does not depend on the position of the chosen point .This post has been edited 2 times. Last edited by parmenides51, Aug 23, 2021, 12:43 AM
Reason: latex
Reason: latex
mean?
![[asy]size(250);import geometry;
pair A=(0,2),B=(-1,0),C=(1,0),P=(1/3,0);
pair M=foot(P,A,B),N=foot(P,A,C);
draw(A--B--C--A);draw(M--P--N);
markrightangle(A,M,P);markrightangle(P,N,A);
pair AA=(0,-2),MM=foot(P,AA,B),NN=foot(P,AA,C);
markrightangle(AA,MM,P);markrightangle(P,NN,AA);
draw(B--AA--C,dotted);draw(MM--P--NN,dotted);
label("$A$",A,dir(90));label("$B$",B,dir(180));label("$C$",C,dir(0));label("$P$",P,dir(90));label("$M$",M,M-P);label("$N$",N,N-P);
label("$A'$",AA,dir(-90));label("$M'$",MM,MM-P);label("$N'$",NN,NN-P);[/asy]](http://latex.artofproblemsolving.com/f/e/3/fe36b0534b924aa764d0c5b1c378227467985a94.png)
be the reflection of 
and 
be the reflections of 
and 
across 
respectively.
,
,
.
is a parallelogram.
and 
,
are collinear. Therefore, 
,
is constant.
