[/quote]
,
prove that![\[ \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{d^2} + \frac{d^2}{a^2}\geq\frac{a}{d} + \frac{b}{a} + \frac{c}{b} + \frac{d}{c}\]](http://latex.artofproblemsolving.com/0/0/4/004e7f148bbb0c797688f5195aa4002dbc316470.png)
If you know same tasks, link them please
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
(
well). i want to study so if you guys have some easy or normal problems please send me
?
chess board, except for the 
central square, there is a checker. In one move, a checker can jump over adjacent checker (diagonally), and this adjacent checker (that was jumped over) is removed. Is it possible to come up with a sequence of moves, so that only one checker is left on the board?
and 
of triangles 
and 
touching their sides 
respectively touch the extension of 
meets 
,
.
and 
is right
with 
and 
.
?
that it is possible to form a triangle from segments of length 
.
times and obtain the numbers 
so that segments with lengths 
and 
Prove that
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
For a positive integer 
,
be the set of bijective functions from 
to itself. For a pair of positive integers 
such that 
,
,
if 
,
?
and a permutation 
?
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
Y by
On the sides 
, 
, 
, 
of the square 
points 
are chosen such that 
The segments 
, 
, 
, 
in the intersection form the quadrilateral 
(see figure). 
a) Prove that is a square.
b) Find the ratio of the areas of the squares and .
(Alexander Shkolny)
, 
, 
, 
of the square 
points 
are chosen such that 
The segments 
, 
, 
, 
in the intersection form the quadrilateral 
(see figure). a) Prove that
is a square.b) Find the ratio of the areas of the squares
and .(Alexander Shkolny)
This post has been edited 1 time. Last edited by parmenides51, Nov 8, 2022, 6:43 PM
