,![\[\frac{1}{a^2+b+1}+\frac{1}{b^2+c+1}+\frac{1}{c^2+a+1}\le1\]](http://latex.artofproblemsolving.com/d/4/3/d431a035ed0b9f699bd232b7c1d0129e659e3309.png)
so that one can split the set 
in to two nonempty disjoint sets 
such that every elements in 
belongs to either 
or 
and there are two elements in the same set whose sum is a cube.
and 
be circles with centres 
and 
,
intersects 
and 
,
lie on 
be the circumcentre of triangle 
.
meets 
and meets 
.
be the orthocentre of triangle 
.
.
Y by
Arthur Morgan is playing a game.
He has
eggs, each with a hardness value 
, where 
is a permutation of the set 
. He is throwing the eggs from an 
-floor building.
When the
-th egg is dropped from the 
-th floor, its new hardness becomes
![\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]](//latex.artofproblemsolving.com/7/5/8/7583af9655a43eb3f6167f955148984a468d4e70.png)
If 
, then the egg breaks and cannot be used again.
Arthur can drop each egg from a particular floor at most once.
For which values of and can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
He has
eggs, each with a hardness value 
, where 
is a permutation of the set 
. He is throwing the eggs from an 
-floor building.When the
-th egg is dropped from the 
-th floor, its new hardness becomes![\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]](http://latex.artofproblemsolving.com/7/5/8/7583af9655a43eb3f6167f955148984a468d4e70.png)
If 
, then the egg breaks and cannot be used again.Arthur can drop each egg from a particular floor at most once.
For which values of
and can Arthur always determine the correct ordering of the eggs according to their initial hardness values?Note: The problem might be wrong or too easy
This post has been edited 1 time. Last edited by JARP091, May 25, 2025, 2:46 PM
Reason: Wrongly LaTeXted
Reason: Wrongly LaTeXted
3, n = 3 is a subproblem that cannot be solved.
1
