AoPS Academy
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.
distinct circles in a plane, with radius 
.
circles, which form a set 
, which satisfy the following., they intersect with each other or, they don't intersect with each other.
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that Prove that
Equality holds iff 
or 
and any cyclic permutations.
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be positive real variables with 
Prove that
Equality holds iff 
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
be non-negative real variables with Prove that
Equality holds iff or 
where 
and 
.
,
and 
be non-negative numbers such that 
.
and set 
The inequality becomes
ưhere
which ends the proofs 