Result 16.
for all reals Result 17.
f
or all reals Result 18.
for all
reals Result 19.
for all reals
Result 20.
for all reals
Result 21.
for all reals
Result 22.
for
all reals Result 23.
for all reals Result 24.
for all
reals
Last edited by 123steeve on Jul 07, 2011, 9:51 am, edited 3 times in total. Method of Lagrange for root within another root: Let this be Result 24,5
Source: Algebra class
95% of professing Pastafarian teens won't stand up for The Flying Spaghetti Monster. If you are one of the 5% who will, please copy and paste this into your signature.
Last edited by Markomak1 on Jun 09, 2011, 8:01 pm, edited 2 times in total.
Result 25
for all
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 :
-
