be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
.
mutually non-attacking rooks placed on a grid 
.
is equal to the largest prime divisor of 
.
be a function differentiable at 0 with 
.
,
,
,
.
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 
of subsets of 
if 
for each 
,
for each 
.
are there? Prove your answer.
Y by Mango247, Mango247, Mango247
Let 
be a square, point 
lies on the side 
, 
be the center of the square, 
be the midpoint of 
, 
. It is known that 
is isosceles. In what ratio does the point divide the side of the square ?.
(Klurman O.)
be a square, point 
lies on the side 
, 
be the center of the square, 
be the midpoint of 
, 
. It is known that 
is isosceles. In what ratio does the point divide the side of the square ?.(Klurman O.)
This post has been edited 3 times. Last edited by parmenides51, Mar 16, 2021, 9:58 PM
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