Berikut ini adalah kumpulan soal mengenai Aljabar tingkat olimpiade. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.
No. 11
Diketahui
x,y\in R,
x\gt2016 dan
x\gt2017. Jika
2016\sqrt{(x+2016)(x-2016)}+2017\sqrt{(x+2017)(x-2017)}=\dfrac12\left(x^2+y^2\right)
maka nilai
xy=
Misal a=\sqrt{(x+2016)(x-2016)}
\(\begin{aligned}
a^2&=x^2-2016^2\\
x^2&=a^2+2016^2
\end{aligned}\)
Misal b=\sqrt{(y+2017)(y-2017)}
\(\begin{aligned}
b^2&=y^2-2017^2\\
y^2&=b^2+2017^2
\end{aligned}\)
\(\begin{aligned}
2016a+2017b&=\dfrac12\left(a^2+2016^2+b^2+2017^2\right)\\
2\cdot2016a+2\cdot2017b&=a^2+2016^2+b^2+2017^2\\
a^2-2\cdot2016a+2016^2+b^2-2\cdot2017b+2017^2&=0\\
(a-2016)^2+(b-2017)^2&=0
\end{aligned}\)
didapat a=2016 dan b=2017
\(\begin{aligned}
x^2&=2016^2+2016^2\\
&=2\cdot2016^2\\
x&=2016\sqrt2
\end{aligned}\)
\(\begin{aligned}
y^2&=2017^2+2017^2\\
&=2\cdot2017^2\\
x&=2017\sqrt2
\end{aligned}\)
\(\begin{aligned}
xy&=2016\sqrt2\cdot2017\sqrt2\\
&=\boxed{\boxed{8132544}}
\end{aligned}\)
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