13.
\( \displaystyle 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{23}=\frac{a}{23!} \),若a除以13的餘數為b,求\( b^3+2b \)除以100的餘數為?
[提示]
a化簡後得23!/13,b=7
已知\( n!=1 \times 2 \times ... \times (n-1) \times n \);若\( \displaystyle \frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...+\frac{11}{12!}=\frac{A}{12!} \),試求A除以10的餘數為何?
(建中通訊解題第62期)
16.p為\( 4x^2+9y^2=36 \)上的動點,若p在第一象限移動,過p點之切線交X軸於A點,交Y軸於B點,O為原點,求\( \overline{OA}+\overline{OB} \)最小值?
[解答]
令p(a,b),\( \displaystyle \frac{a^2}{9}+\frac{b^2}{4}=1 \)
切線\( \displaystyle \frac{a}{9}x+\frac{b}{4}y=1 \),\( \displaystyle \overline{OA}+\overline{OB}=\frac{9}{a}+\frac{4}{b} \)
廣義科西不等式
\( \displaystyle \Bigg[\; \Bigg(\; \frac{a^{\frac{2}{3}}}{9^{\frac{1}{3}}} \Bigg)\; ^3+ \Bigg(\; \frac{b^{\frac{2}{3}}}{4^{\frac{1}{3}}} \Bigg)\; ^3 \Bigg]\;
\Bigg[\; \Bigg(\; \frac{9^{\frac{1}{3}}}{a^{\frac{1}{3}}} \Bigg)\; ^3+ \Bigg(\ \frac{4^{\frac{1}{3}}}{b^{\frac{1}{3}}} \Bigg)\ ^3 \Bigg]\;
\Bigg[\; \Bigg(\; \frac{9^{\frac{1}{3}}}{a^{\frac{1}{3}}} \Bigg)\; ^3+ \Bigg(\ \frac{4^{\frac{1}{3}}}{b^{\frac{1}{3}}} \Bigg)\ ^3 \Bigg]\;
\ge \Bigg(\; 9^{\frac{1}{3}}+4^{\frac{1}{3}} \Bigg)\;^3 \)
17.θ為銳角,\( \displaystyle \frac{16}{sin^6 \theta}+\frac{81}{cos^6 \theta}=625 \),求\( tan \theta \)?
(2005TRML個人賽)
這題可用廣義科西不等式解題
請見
https://math.pro/db/viewthread.php?tid=661&page=1#pid1075