,
to be the least positive integer for which the following holds:
into 
disjoint subsets 
,
.
for which![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](http://latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
so that 
for all 
.
be an acute triangle with orthocenter 
and circumcircle 
.
and 
at 
and 
,
be the circumcenter of 
,
intersects 
.
and the line through 
meet on 
and the domain 
passing through the origin touch 
.
be a mobile point on the parabola 
the distances between the point 
on a coordinate plane such that for all points 
over the domain 
holds.
be the open unit disk in the complex plane and let 
be a holomorphic function such that 
.
be 
.
.
we have the inequality 
,
,
(where the sum has 
is the order of the quotient group 
(the index of the subgroup 
)
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
and 
could be chosen to satisfy 
and 
.
Such That 
is even?
Then 
But I am stuck. I came across this functional equation by trying to find a limit of a sequence, I can show that if it is helpful. 
has solutio
satisfy 
for all 
.
be a fixed circle, and let 
be a point outside 
.
and 
,
is cyclic, where 
are fixed points on the circle.
,
,
is the tangent from 
is the point of tangency.
,
,
is perpendicular to chord 
.
and 
lie on a circle, their cross-ratio is harmonic:![\[
(ANMB) = -1.
\]](http://latex.artofproblemsolving.com/b/d/5/bd5301ba76f493aec0821c454dffc46dea93c0f2.png)
- The intersection points 
,
![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
,
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
that exist function 
such 
and discontinuous on 
Y by
A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules:
i) If the frog jumps one square, it then turns
to the right;
ii) If the frog jumps two squares, it then turns to the left.
Is it possible for the frog to reach the square exactly
squares north of the initial square after some finite number of moves if it is initially facing:
a) North;
b) East?
i) If the frog jumps one square, it then turns
to the right;ii) If the frog jumps two squares, it then turns
to the left.Is it possible for the frog to reach the square exactly
squares north of the initial square after some finite number of moves if it is initially facing:a) North;
b) East?
is such a way that the frog starts at 
and the squares 2024 units north and east are 
and 
respectivly.
with 
.
it faces north, east, west, south respectivly.
times 
squares and then 
squares.
