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be the set of all non-empty sub-strings of "Socrates".
but "teS", "sorat", "setarcoS" 
.
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be . Then the sum of these two strings will be the string formed by all the letters between the first term of and the last term of . For instance, "Socr" + "te" 
"Socrate" and "cra" + "ates" "crates".
such that
.
is still the same.
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.
such that 
. Prove that
be a positive integer, and consider the equation:
Answer the following questions. You may assume that 
is known.
Determine the number of real solutions of equation 
for each .
For 
, let 
be the segond largest real solution of .
Find 
such that 
Find 
, where 
is defined as
has property 
if :![\[
\begin{cases}
\text{the set } A \text{ has at least } 4 \text{ elements} \\
\text{the element } 1+1 \text{ is invertible}\\
x+x^4=x^2+x^3 \text{ holds for all } x \in A
\end{cases}
\]](http://latex.artofproblemsolving.com/b/6/9/b6993ee76eb3792e08d441f5d5c463247c392c4a.png)
a) Prove that if a ring has property , and 
are distinct elements, such that 
and 
are units, then 
is also a unit, but 
is not a unit..
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?
and 
: Prove that :
,
are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles.
