chứng minh bất đẳng thức và phân tích đa thức thành nhân tử ( đề thi hsg)

1/ CMR: nếu a,b,c là 3 cạnh của tam giác thì A > 0 với
2a^2b^2 + 2b^2c^2 + 2a^2c^2 - a^4 - b^4 - c^4
2/ CMR a/(b+c-a) + b/(c+a-b) + c/(a+b-c) >hoặc= 3
3/ phân tích đa thức thành nhân tử
a) (x-y)^3 + (y-z)^3 + (z-x)^3
b) a^2(b-c) +b^2(c-a) +c^2(a-b)
c)cho a+b+c+d=0 . CMR : a^3 +b^3+c^3+d^3=3(c+d)(ab-dc)
d) (a+b+c)^3-(a+b-c)^3 +(b+c-a)^2 - (c+a-b)^2
thông cảm mình ko biết đánh latex . mấy bn help mình mấy bt này nha

 

Câu 1:

$A= 2a^2b^2 + 2b^2c^2+ 2a^2c^2 - a^4 - b^4 - c^4$

$\iff A= 4a^2c^2-(a^4 +b^4 +c^4 - 2a^2b^2+ 2a^2c^2- 2b^2c^2)$

$\iff A= 4a^2c^2 - (a^2 -b^2 +c^2)^2$

$\iff A= (2ac+a^2 - b^2+c^2)(2ac - a^2 +b^2 - c^2)$

$\iff A= ((a+c)^2-b^2)(b^2-(a-c)^2)$

$\iff A= (a+b+c)(a+c-b)(b+a-c)(b-a+c)$

Mà $a,\ b,\ c$ là $3$ cạnh của tam giác nên:$\begin{cases}a+b+c > 0\\a+c-b > 0\\b+a-c >0\\b-a+c >0\end{cases} \Longrightarrow (a+b+c)(a+c-b)(b+a-c)(b-a+c)>0$

$\Longrightarrow A > 0\ (\mathfrak{Dpcm})$

 

Bài 2:

Ta có bđt phụ: $(a+b-c)(b+c-a)(c+a-b) \le abc$

Ta có:

$\dfrac{a}{b+c-a} + \dfrac{b}{c+a-b} + \dfrac{c}{a+b-c} \ge 3\sqrt[3]{\dfrac{abc}{((a+b-c)(b+c-a)(c+a-b)}} \ge 3\sqrt[3]{\dfrac{abc}{abc}} \ge 3$

 

Câu 1:

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⟺A=4a2c2−(a4+b4+c4−2a2b2+2a2c2−2b2c2)" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟺A=4a2c2−(a4+b4+c4−2a2b2+2a2c2−2b2c2)⟺A=4a2c2−(a4+b4+c4−2a2b2+2a2c2−2b2c2)

⟺A=4a2c2−(a2−b2+c2)2" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟺A=4a2c2−(a2−b2+c2)2⟺A=4a2c2−(a2−b2+c2)2

⟺A=(2ac+a2−b2+c2)(2ac−a2+b2−c2)" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟺A=(2ac+a2−b2+c2)(2ac−a2+b2−c2)⟺A=(2ac+a2−b2+c2)(2ac−a2+b2−c2)

⟺A=((a+c)2−b2)(b2−(a−c)2)" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟺A=((a+c)2−b2)(b2−(a−c)2)⟺A=((a+c)2−b2)(b2−(a−c)2)

⟺A=(a+b+c)(a+c−b)(b+a−c)(b−a+c)" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟺A=(a+b+c)(a+c−b)(b+a−c)(b−a+c)⟺A=(a+b+c)(a+c−b)(b+a−c)(b−a+c)

Mà a, b, c" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">a, b, ca, b, c là 3" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">33 cạnh của tam giác nên:{a+b+c>0a+c−b>0b+a−c>0b−a+c>0⟹(a+b+c)(a+c−b)(b+a−c)(b−a+c)>0" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⎧⎩⎨⎪⎪⎪⎪a+b+c>0a+c−b>0b+a−c>0b−a+c>0⟹(a+b+c)(a+c−b)(b+a−c)(b−a+c)>0{a+b+c>0a+c−b>0b+a−c>0b−a+c>0⟹(a+b+c)(a+c−b)(b+a−c)(b−a+c)>0

⟹A>0 (Dpcm)" role="presentation" style="font-family: "Open Sans", sans-serif; display: inline; line-height: normal; font-size: 15px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⟹A>0 (Dpcm)