and 
be 2 distinct positive integers.
be in an integer. Prove why ?
with maximums 
and 
.
,
can be expressed as 
where 
is a positive integer, find 
and 
.
Let 
.
Let 
.
Let 
.
Y by Math-Ninja, Williamgolly, van1sh, Gestapo, TwinPrime, mathleticguyyy, Vietjung, Toinfinity, Lol_man000, OlympusHero, tsarnicky123, samrocksnature, Adventure10
Let 
and 
. Find the minimum value of 
and 
. Find the minimum value of 
![$\frac{a^4+b^4}{a^3+b^3}+\frac{b^4+c^4}{b^3+c^3}+\frac{c^4+a^4}{c^3+a^3}\ge a+b+c \ge3\sqrt[3]{abc}=3$](http://latex.artofproblemsolving.com/2/7/6/2761477d952109a0bb7f6fe4a336a89d2b1b0ded.png)
Equality iff 
be positive real numbers.Prove that 
be positive real numbers.Prove that 
equality iff 
equality iff 
.
and 
.
Expand the left side, then use AM-GM to prove the inequality
which is true (all terms on left are square of right)
be positive real numbers such that 
and 
Prove that
Note by AM-GM that
as desired. (The inequality is strict because the equality case 
is not possible as given in the problem statement.)
and 
be positive numbers such that 
Prove that
,
and replace 
by a new 
by a new 
.
which, due to both sides being positive, is equivalent to squaring both sides
which in turn is equivalent to its both sides squared again then the terms collected and factored
QED
be positive numbers such that 
Prove that
is convex on 
.
QED
as suggested by the attachment to the above post?
The same procedure applies to the term involving 
and 
Prove that
be positive reals. Prove that
and 
Prove that
be positive numbers such that 
Prove that 
be positive numbers such that ![\[ \overset{n}{\underset{i=1}{\sum}} \frac{1}{a_i+n-1} \le 1\]](http://latex.artofproblemsolving.com/d/6/b/d6b71e62d8728548b4dce109e9dac1a9b5e29b77.png)
Prove that
