几十个十分有用的代数恒等式
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⼏⼗个⼗分有⽤的代数恒等式
6h404216p3682601
THE-BEAUTY-INSIDE-OF-INEQUALITY
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80条最有⽤的代数变形
三元恒等式(整理中)艾术
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2015年全国⾼中数学联赛讲义之第九讲：常⽤恒等式与常⽤不等式If , then
If ,then
By Ferrari's Identity we have:
If be positive real numbers，then
.
where comes from the Sophie Germain identity. ,
,
.
.
.
:
Let , , be non-negative real numbers for which . Prove that Result 1. - Difference of same powers
for non-negative integers and all reals .
Result 2. - Sum of same odd powers
for non-negative integers and all reals .
Result 3. - Special Case of Result 1. - Difference of squares for all reals .
Result 4. - Special Case of Result 1. - Difference of cubes for all reals .
Result 5. - Sum of Squares - Different representations
for all reals .
Result 6.
for all reals .
Result 7.
for all reals .
Result 8.
for all reals .
Result 9.
for all reals .
Result 10.
for all reals .
Result 11.
for all reals .
Result 12.
for all reals .
Result 13.
for all reals
Result 14.
for all reals
Result 15.
for all reals
Result 16.
for all reals
Result 17.
for 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
Result 24,5
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 12387198739136518 742687632 problems, who knows? I just saw and decided to generalize it.
Result31:
Result32
Result 33
-from Problems from the Book
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: "Lagrange's Identity"
for all reals
Result 40.
, for .
Result 41.
for if and only if one or both of the following conditions are true: .
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. Result 44.
Result 45.
Result 56.
Result 58
Result 59
Result 60
Result 61
A useful substitution
Result 62
Result 63
Result 64
Result 65
Result 66
Result 67 :
-
-
Result 68
Result 69
Result 70:
whre
Result 71:
If or then
Result 71:
Result 72:
Result 68: then .
.
If then
.。