AoPS Academy
. Then number of real roots of 
, is circumscribed around the balls. Then equals
. The center 
of the semi-circle is the midpoint of segment 
. We have 
and 
..
be a function differentiable at 0 with 
. Find
, 
, 
, 
.
and its absolute and relative error, known that its absolute error is equal or lower than 
If G has a normal subgroup of order 
then prove that 
is abelian without using Sylow Theorems.
![\[
\int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx
\]](http://latex.artofproblemsolving.com/e/a/9/ea9f3a093b1d4c0fa39e5b6d42afa2f71dace1e0.png)
a prime number, with 
and 
, 
vector space.
is a group?
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
has no real roots, and 
takes all real values 
,
must all be 
.
,
,
--- and so