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real Result 26

for all real

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for all real

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for all real Result 30

tbh I don't really know how that could be remotely useful. You might see it once in like every 12387198739136518742687632 problems, who knows? I just sawResult 31

Not sure if this one deserves to be called a \#24. Due to the \be more applicable to, say, if you were trying to solve a diophantine equation via factoring, where you could just \necessary \

#31 was from a problem by Titu Andreescu:Find all pairs (x,y) of integers such thatResult 32.

and decided to generalize it.

(Titu Andreescu)

Result 33

Result 34

-Sophie Germain

What about substitutions? Like: Result 35

Result 36

for all

reals Result 37

.

for all reals

Result 38

.

for all reals .

Result 39: \

for all reals

Result 40.

, for

Result 41.

for

true:

.

.

if and only if one or both of the following conditions are

Pages 26-29 of Secrets in Inequalities - Volume 2 - Advanced Inequalities by Pham Kim Hung, published by GIL, has some great

factorizations. I might post them sometime, but not before I am comfortable with all of them. Anyone else may feel free to post them, since you can't put a copyright on mathematical identities... I think. Some Substitutions: Result 42.

for

Result 43. If

are the sides of a triangle, the identity or inequality can be transformed

.

to , where . The converse is also true.

(This is somewhat related to geometry, but it allows one to brute-force proof a geometric inequality with algebraic technniques that hold for postiive reals instead of the inconvenient sides of a triangle). Result 44.

Result 45.

Should be:

Result31:

Moderator Edit: Result 32.

Should be:

Result32

Result 56.

Result 57. The maximum of

is

Result 58

.

Result 59

Result 60

Result 61

A useful substitution

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Result 63

Result 64

Result 65

Result 66

Result 67 :

-