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be a nonempty set of positive integers. We say that a positive integer 
is clean if it has a unique representation as a sum of an odd number of distinct elements from . Prove that there exist infinitely many positive integers that are not clean.
such that for any positive integer 
, satisfies 
.
where 
denotes the Fibonacci sequence.**
be a sequence such that 
and 
.
and 
)
is a derivable function with the property 
, for every 
, then f is a polynomial function of degree smaller or equal than 1
be a function that admits a primitive 
such that 
if 
such that 
be holomorphic functions on 
, where is an open, connected subset of 
. Suppose that the function 
given by 
takes a maximum value in . Must each function 
be constant on ?
defined on 
by![\[f(x, y) = x^4 + y^4 - 2(x - y)^2.\]](http://latex.artofproblemsolving.com/1/d/9/1d9e85722971f67e3ba2dee0e12d119ceabc671a.png)
(and determine them) such that![\[\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,\]](http://latex.artofproblemsolving.com/e/b/3/eb384916db4e019c7ba11f52fadf973f515891e2.png)
where 
denotes the Euclidean norm.
and 
and there exists a positive integer such that 
. Let 
be minimal polynomial 
over 
. Prove that 
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
![$\frac{d}{dx} [\int_0^{e^x} \log (t) \cos ^4(t) dt]$](http://latex.artofproblemsolving.com/c/c/5/cc57ad163c659a09cb0cf4c3dda9198ac398cfd3.png)
.
.
.
.
.
if we put 
get 
.
is decreasing in
and increasing in
.
has local minimum at .and no local Maxima but has global maximum.
Compute - For define Determine the open interval(s) (if any) where is decreasing and all the open interval(s) (if any) where is increasing. Determine all the local minima of (if any) and all the local maxima of (if any) 
by newton leibntiz (which is nothing but an application of FTC) we get, ![$\frac{d}{dx}\left[\int_0^{e^x}\log(t)\cos^4(t)dt\right]=e^x(x\cos^4(e^{x}))$](http://latex.artofproblemsolving.com/b/5/0/b50f25dff4ee175490c62f92c8b7a8ea5a9a6413.png)
so decreasing in the interval 
and increasing in 
vanishes only at and we observe that 
hence it has a local minima at 
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
dec on 
and inc on 
is a local minima (easy to see from above)