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צפה בגרסה המלאה : שאלות קשות באולמפידה



מומד
06-11-2013, 14:41
מהו פתרון השאלות הזו

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antigravity
05-09-2014, 22:42
נסיון לפתור את 10:
נתון \frac{ad+bc}{bd}+\frac{ab+cd}{ac}=6.
נקרא לביטוי שרוצים למקסם \varphi. ניתן לרשום \varphi=\frac{ad+bc}{cd}+\frac{ad+bc}{ab}.
כלומר \varphi=(ad+bc)(\frac{1}{cd}+\frac{1}{ab})=(ad+bc) (\frac{ab+cd}{abcd})=(6-\frac{ab+cd}{ac}) \frac{ab+cd}{ac} כאשר המעבר האחרון לפי הנתון.
ידוע כי נק' המקסימום של הפונ' f(x)=(6-x)x היא ב-x=3 ולכן הערך המקסימלי של \varphi הוא (6-3)3=9. (זאת בהנחה שקיימים ממשיים a,b,c,d\neq0 כך ש-\frac{ab+cd}{ac}=3, אבל זה כמובן נכון.)