of them there were either 
who were all friends among each other or 
and 
play a game, given an integer 
,
first, then every player sees the last number written and if it is 
then in his turn he writes 
or 
,
be the circumcircle of a scalene triangle 
.
be a circle internally tangent to 
and 
,
.
touch 
and 
,
and 
concur on the bisector of 
.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
,
?
with edge 
cm.
and tangent to the plane 
.
)
and a plane 
,
and 
.
.
and 
,
and 
.
.
be the midpoint 
.
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
,
and 
are all irrational numbers.
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
Y by
Lets define Graph G as a graph with "n" vortexes and no edges. Define C(G) as a number of cycles that starts from a point, visit all points exactly once, and comes back to the point that they started (The paths made can't cross each other). Define R(G) as a number of routes that starts from a point, visit all points exactly once, and finishes at another point. (The paths made cannot cross each other.) Show that R(G)≥n*C(G). Also, show the reason why is it inequality instead of an equation, and show the equrilibrum conditions.
(See attachment for example)
(See attachment for example)
Attachments:
This post has been edited 1 time. Last edited by jaydenkaka, Oct 24, 2024, 9:44 AM
