Ketaksamaan SEAMO 2005


Misalkan a,b,c dan d bilangan real positif. Tunjukan bahwa \frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4} \ge \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}

SOLUSI

Menurut AM-GM. diperoleh

1+\frac{b^4}{c^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4} \ge \frac{b}{a}

1+\frac{a^4}{b^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4} \ge \frac{c}{b}

1+\frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{d^4}{a^4} \ge \frac{d}{c}

1+\frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{c^4}{d^4} \ge \frac{a}{d}

\frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4} \ge4

Jumlahkan ketaksamaan-ketaksamaan di atas, kita peroleh

4+4(\frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4}) \ge4+4(\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d})

Tentunya ekuivalen dengan

\frac{a^4}{b^4}+\frac{b^4}{c^4}+\frac{c^4}{d^4}+\frac{d^4}{a^4} \ge \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d})

Terbukti sudah.

About ardiantoarsadi

don't look for miracles it will come

Posted on Januari 24, 2010, in SOAL DAN SOLUSI and tagged , . Bookmark the permalink. 1 Komentar.

  1. masih sma kelas 1 ya? hebat banget

Tinggalkan Balasan

Isikan data di bawah atau klik salah satu ikon untuk log in:

Logo WordPress.com

You are commenting using your WordPress.com account. Logout / Ubah )

Gambar Twitter

You are commenting using your Twitter account. Logout / Ubah )

Foto Facebook

You are commenting using your Facebook account. Logout / Ubah )

Foto Google+

You are commenting using your Google+ account. Logout / Ubah )

Connecting to %s

%d blogger menyukai ini: