[/quote]
,
is an integer. Prove that 
are integers.
,
be the foot of the perpendicular from 
to line 
.
lies inside triangle 
and 
.
meets the circle with diameter 
.
such that there exist integers 
so that the polynomial 
has 4 distinct roots in the interval 
?
?
in natural numbers m and n.
.
such that ![$gcd(p[i], p[i+1])=1$](http://latex.artofproblemsolving.com/f/b/b/fbb304fb8ebdb8987edc830234698de2dc7a28a2.png)
(here ![$p[i]$](http://latex.artofproblemsolving.com/0/b/6/0b6cc7e1108ab6b64f3a644a3b28891f826b3bea.png)
refers to the 
be not cyclic and haves edges not parallel to each other.
as the intersection of the tangent line at 
with respect to circle 
and the 
-
and 
as the intersection of lines 
and 
,
,
,
,
are collinear.
(
well). i want to study so if you guys have some easy or normal problems please send me
and 
.
.
of degree ![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of ![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
and 
Prove that
Let 
and 
Prove that
not isosceles, the incircle 
.
is the intersection of 
.
is the projection of 
.
cut 
is the projection of 
.
cut 
respectively. Prove 
are collinear
Y by
Let 
be a positive integer. Find the smallest positive real 
, satisfying the following condition: if 
is a connected graph with 
vertices and 
edges, then it is always possible to delete at most 
edges, so that the resulting graph has a proper vertex coloring with two colors.
be a positive integer. Find the smallest positive real 
, satisfying the following condition: if 
is a connected graph with 
vertices and 
edges, then it is always possible to delete at most 
edges, so that the resulting graph has a proper vertex coloring with two colors.
.
for some 
- just take the complete graph and see we cannot make 
.
.
.
