AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
, 
, 
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
and 
be 2 distinct positive integers.
be in an integer. Prove why ?
of integers, a pair 
with 
is called interesting if there exists a pair 
of integers with 
such that 
For each 
, find the largest possible number of interesting pairs in a sequence of length 
.
such that, for all 
,
cards on which the numbers 
, 
, 
, are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let 
be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer 
for which Bob can guarantee that is at least .
be positive real number such that 
Find the minimum value of 
other than a polynomial satisfying
prove inequality 
. 
Prove that
Where 
![\[\begin{cases}
3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\
2x+\dfrac{1}{x-y}=4
\end{cases}\]](http://latex.artofproblemsolving.com/e/b/f/ebf41764f74c98da66dc31c48975f27bd734c334.png)
be a triangle where 
(the constant 
) .Prove that
are interior angles of a triangle, the condition 
from which it follows that
where 
,
,
and 
with 
.
,
,
thereby
(the constant 
from which it follows that
Since 
,
,
when 
;
when 
,
.
is 
,
and 
,
,
.
.
) .Then
(the constant 
(the constant 
from which it follows that
Since that 
thereby 
,
.
has no maximum. 
from which it follows that
where 
and 
with 
.
.
is acute. This means that 
.
.
(the constant 
(the constant 
Where 
Where 
Where 
Prove that
Where 
and 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Prove that 
Let 
Prove that 
.![$abc\geq 3\sqrt[3]{abc}+2$](http://latex.artofproblemsolving.com/7/a/b/7ab2f6a8d1099cb193ad35dfce08897d7eb25f33.png)
.
,
which is equivalent to 
which asserts 
or 
.![$(ab+bc+ca)^{2}\geq 3abc(a+b+c)\geq 9abc\sqrt[3]{abc}=144$](http://latex.artofproblemsolving.com/a/2/c/a2c95a4971acdf865fcaf77902f9242f512ba420.png)
.
![$$ab+bc+ca\geq 3\sqrt[3]{(abc)^2}\geq12$$](http://latex.artofproblemsolving.com/2/c/9/2c988b04ff2bc784b81538dd2cbe02628a7cfd41.png)
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
and 
Prove that
such that 
Prove that
be non-negative numbers such that 
In 
. Prove that
and 
Prove that
and 
prove that:
Ukraine2008？
Prove that
Let 
P
Prove that
we are done!
Let 
,
,
be positive real numbers such that 
.
Let 
be positive real numbers such that 
![\[\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/b/69b62a5866ba96e43b0c59f89daae9710a896c45.png)
.Prove that
,
,
by Muirhead 
.
be positive real numbers . Prove that
Where 
be positive real numbers such that 
Prove that
