1.
\( \displaystyle 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}} \)
4.
設ABCD是邊長為1的正方形,P點在\( \overline{BC} \)上,Q點在\( \overline{CD} \)上,且\( ∠PAQ=45^o \),試證\( \displaystyle \frac{\overline{AB}+\overline{BP}}{\overline{AD}+\overline{DQ}}=\frac{\overline{AP}^2}{\overline{AQ}^2} \)
正方形ABCD的\( \overline{BC} \),\( \overline{CD} \)邊上各有一點M,N,若\( ∠MAN=45^o \),試證:\( \displaystyle \frac{\overline{AM}}{\overline{AN}}=\sqrt{\frac{\overline{AB}+\overline{BM}}{\overline{AD}+\overline{DN}}} \)
(建中通訊解題 第23期)
5.
\( \displaystyle \lim_{n \to \infty} \cdot \lim_{n \to \infty} \Bigg(\; \frac{1+\root n \of {1^n+2^n}+\root n \of{2^n+3^n}+...+\root n \of{(m-1)^n+m^n}}{m^2} \Bigg)\; \)
(高中數學101 P373)
6.
\( \displaystyle cos \frac{\pi}{7}-cos \frac{2 \pi}{7}+cos \frac{3 \pi}{7} \)
(IMO 1963,
http://www.imo-official.org/year_info.aspx?year=1963)
化簡\( \displaystyle cos \frac{6 \pi}{7}-cos \frac{5 \pi}{7}+cos \frac{4 \pi}{7} \)的值為?
(100全國高中聯招,
https://math.pro/db/thread-1163-1-1.html)