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that satisfy 
, the factorial 
is defined as a product of all integers between 
and inclusive: 
.
denote the sum of the first 
factorials, i.e.
Find the remainder when 
is divided by 
such that 
and 
.
be a positive integer, and 
be two complex numbers such that 
and 
, for any 
. The matrices 
satisfy the relation 
. Prove that 
and 
are invertible.
is solvable over 
for every 
. Here 
, 
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center and radius 
, for some 
. Prove that ![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
. be a positive integer and ![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree . Prove that ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and 
. Prove that ![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
be an odd prime number, and be an odd number not divisible by . Consider a field 
be a field with 
elements, and 
be the set of elements of 
, whose order is not in the multiplicative group 
. Prove that the polynomial 
has at least coefficients equal to .
and 
be a continuous function for which there exists an antiderivative 
, such that 
, for any 
, and
holds for any 
. Prove that 
for all .
be a positive integer, ![$g \in \mathbb{R}[X]$](http://latex.artofproblemsolving.com/0/1/e/01e7ebcebcb9f16935716ee859f7acfaa66c29d4.png)
, 
be a polynomial with all of its roots being real, and a polynomial function such that 
for any . Prove that 
for all .
be two matrices such that 
. Prove that: is odd, then 
;
, then .
matrices which has only 
and for their entries?
, with their second derivative being continuous, such that the following holds for all 
: ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
and 
. Then we should find this ratio: 
.![\[S_{EGC}=(a+x)^2+\frac{x^2}{2}-\frac{(a+x)(a+2x)}{2}-\frac{a(a+x)}{2}=\frac{x^2}{2}.\]](http://latex.artofproblemsolving.com/2/a/7/2a7271b3023b31cbdadac9161187b92f1a292986.png)
![\[S_{DEFBC}=(a+x)^2+x^2+\frac{ax}{2}.\]](http://latex.artofproblemsolving.com/c/f/0/cf022939edf4bba3c1ce126b096a1e193665deee.png)
![\[S_{DEFBC}=16S_{EGC}\Rightarrow a^2+2.5ax-6x^2=0\]](http://latex.artofproblemsolving.com/e/a/6/ea6e9085f8aac29b7c60539cc726c5be3c33a48e.png)
![\[\left(\frac{a}{x}\right)^2+2.5\frac{a}{x}-6=0,\quad \frac{a}{x}>0\Rightarrow\frac{a}{x}=\frac{3}{2}.\]](http://latex.artofproblemsolving.com/f/3/0/f3047f7e354ad1e790923424d1205422532724df.png)
Therefore,![\[\left(\dfrac{x}{a+x}\right)^2=\left(\dfrac{1}{\frac{a}{x}+1}\right)^2=\frac{4}{25}.\]](http://latex.artofproblemsolving.com/9/3/8/9388ead9972342deafa8461cdcb8961e91254ed0.png)
Hence answer is: 
.
lies on side 
of square 
and square 
is drawn outwards 
is 
of the area of pentagon 
.
