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.
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 .
such that
for any positive integers 
, 
.
, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.
and 
positive real numbers. Prove that![\[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]](http://latex.artofproblemsolving.com/e/9/e/e9e73a14f4dfa6315e7562998a9394d371847139.png)
and 
Prove that
Let 
and Prove that
Let 
and Prove that
Let 
and 
Show that![\[\sum_{k=1}^n\frac{1}{1+\sum_{j\neq k}x_j}\le\frac n{1+(n-1)\sqrt[n]{x_1x_2\cdots x_n}}.\]](http://latex.artofproblemsolving.com/e/4/c/e4ca07e5003fad8937af3c44dcbc9906fa48e5b4.png)
with 
the following inequality holds:
, 
, 
, the inequality![\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]](http://latex.artofproblemsolving.com/f/a/9/fa964e29de91f0e2f3cb32e335a52d4521bc38fd.png)
, , be positive real numbers such that 
. Prove that![\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.
\]](http://latex.artofproblemsolving.com/b/2/9/b295b0b2a765e41a639657e13b724c60323fa208.png)
where 
is in radians.
be distinct reals such that 
Prove that 
be the altitude and 
be the intersection point of the altitudes of triangle 
. Point is symmetric to with respect to 
. The circumcircle of triangle 
intersects at points 
and 
. Prove that 
.
Prove that
Prove that
