[/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.
integers where 
.
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
On an infinite sheet of tiles, an infinite number of 
tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with checkered rectangles?
tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with checkered rectangles?
