[/quote]
,
such that 
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
be a set of Natural numbers from 
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the midpoint of 
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
)
be an acute triangle with 
,
be the projection from 
.
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
Y by
Let 
be a given integer, 
. Consider a graph 
with 
vertices satisfying the condition: for any two non-adjacent vertices 
and 
in graph , the sum of their degrees must satisfy 
. Please answer the following questions and prove your conclusions.
(1) Suppose the length of the longest path in graph is 
satisfying the inequality 
, does graph necessarily contain a cycle of length 
? (The length of a path or cycle refers to the number of edges that make up the path or cycle.)
(2) For the case where
and graph is connected, can we determine that the length of the longest path in graph , 
?
(3) For the case where
, is it necessary for graph to have a path of length 
(i.e., a Hamiltonian path)?
be a given integer, 
. Consider a graph 
with 
vertices satisfying the condition: for any two non-adjacent vertices 
and 
in graph , the sum of their degrees must satisfy 
. Please answer the following questions and prove your conclusions.(1) Suppose the length of the longest path in graph
is 
satisfying the inequality 
, does graph necessarily contain a cycle of length 
? (The length of a path or cycle refers to the number of edges that make up the path or cycle.)(2) For the case where
and graph is connected, can we determine that the length of the longest path in graph , 
?(3) For the case where
, is it necessary for graph to have a path of length 
(i.e., a Hamiltonian path)?
