real Result 26
for all real
Result 27
for all real
Result 28
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
Result 62
Result 63
Result 64
Result 65
Result 66
Result 67 :
-