[/quote]
,
knights sitting at a round table. They consist of 
be a cyclic quadrilateral, 
,
,
,
,
and 
Prove that:
,
satisfy 
.
where 
.
and other 
being 0.)
such that 
,
.
be a triangle such that 
.
,
at points 
,
,
be its centre. Let 
be a point on the side 
and 
intersect for the second time at 
.
be the foot of the perpendicular from 
to the line 
.
,
beyond 
,
beyond 
.
be a convex pentagon with 
and 
.
and 
.
if and only if area 
+ area 
= area 
+ area 
.
and circumcenter 
,
.
be the midpoint of 
and 
intersects 
and 
intersects 
and 
at 
and 
,
wrt 
is tangent to the circumcircle of 
.
be the set of all lines in the plane and let 
be a function that assigns to each line 
a point 
on 
.
,
passing through 
,
for any line 
with 
,
is a tree with 
edges. Prove that the 
is the circumcircle of 
,
is an arbitrary point on 
at 
is the reflection point of 
.
is parallel to 
is parallel to 
is the circumcenter of 
is a pair of isogonal lines wrt 
is given. Prove that there exists a sequence 
of rational numbers with the following properties:
;
,
or 
;
is an integer for some 
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
