be the set of positive integers. Find all functions 
such that, for all 
:
balls, with 
red, 
be complex numbers such that [center]
for any 
with 
[/center]
and 
for any 
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
.
be a point on the arc 
(not containing 
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
edges of a regular icosahedron are distinguished by labeling them 
How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
.
and their entries belong to the set 
.
For all 
be a polynomial with integer coefficients. Define a sequence 
of integers such that 
and 
for all 
.
for which 
then either 
or 
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center 
,
.![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
.![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
be the set of the differentiable functions 
satisfying 
for any 
then the set of solutions of the equation 
is infinite.
given by 
Show that 
exists everywhere and find the direction along which the absolute value of the directional derivative is maximized at the point 
in coordinates, then along any direction given by the unit vector 
the directional derivative of 
at the point 
is 
and 
be integers and 
a square matrix of order 
there is a function 
such that replacing the entries 
of 
respectively by 
the determinant of the matrix 
thus obtained is not divisible by 
a real number. Given a differentiable function 
,
the curve 
on the XY-plane. For all point 
and 
.
belongs to 
two non-zero matrices such that 
and 
.
Y by
A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer 
with another positive integer 
, as long as 
is a perfect square. For example, if the number on the blackboard is 
, Carmela can replace it with 
, because 
, then replace it with 
, because 
. If the number on the blackboard is initially 
, determine all integers that Carmela can write on the blackboard after finitely many operations.
with another positive integer 
, as long as 
is a perfect square. For example, if the number on the blackboard is 
, Carmela can replace it with 
, because 
, then replace it with 
, because 
. If the number on the blackboard is initially 
, determine all integers that Carmela can write on the blackboard after finitely many operations.
can be written after finitely many operations.
can.
,
and 
.
,
.
with 
,
,
.
,
.
can be written, then since 
,
can be written then 
can be written.
can be written at some point.
