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Tính $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}$

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#1
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Cho $\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=-2$ và $\frac{a}{x}-\frac{b}{y}-\frac{c}{z}=0$. Tính $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}$

$\large{\int_{0}^{\infty }xdx<\heartsuit}$

#2
Zaraki

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Cho $\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=-2$ và $\frac{a}{x}-\frac{b}{y}-\frac{c}{z}=0$. Tính $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}$


Ta có $$\begin{aligned} \frac xa - \frac yb - \frac zc =-2 & \Rightarrow \left( \frac xa - \frac yb - \frac zx \right)^2=4 \\ & \Leftrightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}+ \frac{z^2}{x^2}+ 2 \cdot \left( \frac{yz}{bc}- \frac{xy}{ab}- \frac{zx}{ac} \right)=4 \\ & \Leftrightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}+ \frac{z^2}{x^2}+ 2 \cdot \dfrac{abc(ayz-cxy-bzx)}{a^2b^2c^2}=4 \\ & \Leftrightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}+ \frac{z^2}{x^2}+ 2 \cdot \dfrac{ayz-cxy-bzx}{abc}=4 \end{aligned}$$
Lại có $\frac ax- \frac by - \frac cz =0 \Leftrightarrow \frac{ayz-bzx-cxy}{xyz}=0 \Leftrightarrow ayz-bzx-cxy=0$. Do đó $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}+ \frac{z^2}{x^2}= \boxed{4}.$$

Discovery is a child’s privilege. I mean the small child, the child who is not afraid to be wrong, to look silly, to not be serious, and to act differently from everyone else. He is also not afraid that the things he is interested in are in bad taste or turn out to be different from his expectations, from what they should be, or rather he is not afraid of what they actually are. He ignores the silent and flawless consensus that is part of the air we breathe – the consensus of all the people who are, or are reputed to be, reasonable.

 

Grothendieck, Récoltes et Semailles (“Crops and Seeds”). 


#3
Zaraki

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Tổng quát: Cho $\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ \cdots + \frac{a_n}{b_n}=k$ với $n \in \mathbb{N}^*$. và $\frac{b_1}{a_1}+ \frac{b_2}{a_2}+ \cdots + \frac{b_n}{a_n}=0$. Chứng minh $$\frac{a_1^2}{b_1^2}+ \frac{a_2^2}{b_2^2}+ \cdots + \frac{a_n^2}{b_n^2}=k^2$$
Mọi người thử chứng minh coi có đúng không nhỉ :wub:

Bài viết đã được chỉnh sửa nội dung bởi Phạm Quang Toàn: 27-12-2012 - 16:34

Discovery is a child’s privilege. I mean the small child, the child who is not afraid to be wrong, to look silly, to not be serious, and to act differently from everyone else. He is also not afraid that the things he is interested in are in bad taste or turn out to be different from his expectations, from what they should be, or rather he is not afraid of what they actually are. He ignores the silent and flawless consensus that is part of the air we breathe – the consensus of all the people who are, or are reputed to be, reasonable.

 

Grothendieck, Récoltes et Semailles (“Crops and Seeds”). 





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