Olimpiade Zone : Aljabar [2]

Berikut ini adalah kumpulan soal mengenai Aljabar tingkat olimpiade. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

• 1
• 2

---

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 $xy=$

SUMBER

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 $b=2017$

$\begin{aligned}x^2&=2016^2+2016^2\\&=2\cdot2016^2\\x&=2016\sqrt2\end{aligned}$

$\begin{aligned}y^2&=2017^2+2017^2\\&=2\cdot2017^2\\x&=2017\sqrt2\end{aligned}$

$\begin{aligned}xy&=2016\sqrt2\cdot2017\sqrt2\\&=\boxed{\boxed{8132544}}\end{aligned}$

Jadi, $xy=8132544$.

• 1
• 2

Share this post