2.
已知正數\(a,b\)滿足條件\(log_9 a=log_{12}b=log_{16}(a+b)\)，則\(\displaystyle \frac{b}{a}\)之值為何？
(A)\(\displaystyle \frac{4}{3}\) (B)\(\displaystyle \frac{8}{5}\) (C)\(\displaystyle \frac{1+\sqrt{3}}{2}\) (D)\(\displaystyle \frac{\sqrt{5}-1}{2}\) (E)\(\displaystyle \frac{1+\sqrt{5}}{2}\)
設\( p,q \in R \)且\( p>0,q>0 \)，若\( log_9 p=log_{12}q=log_{16}(p+q) \)，則\( \displaystyle \frac{q}{p} \)之值介於下列哪一各區間？
(A) \( \displaystyle (1,\frac{3}{2}) \) (B) \( \displaystyle ( \frac{3}{2},2) \) (C) \( \displaystyle (2,\frac{5}{2}) \) (D) \( \displaystyle ( \frac{5}{2},3 ) \) (E) \( \displaystyle ( 3,\frac{7}{2} ) \)
(100彰化藝術高中,田中高中，
https://math.pro/db/viewthread.php?tid=1152&page=1#pid3661)
Suppose that \(p\) and \(q\) are positive numbers for which\(log_{9}p=log_{12}q=log_{16}(p+q)\).What is the value of \(\displaystyle \frac{q}{p}\)?
(A)\(\displaystyle \frac{4}{3}\) (B)\(\displaystyle \frac{1+\sqrt{3}}{2}\) (C)\(\displaystyle \frac{8}{5}\) (D)\(\displaystyle \frac{1+\sqrt{5}}{2}\) (E)\(\displaystyle \frac{16}{9}\)
(1988AHSME，
https://artofproblemsolving.com/ ... Problems/Problem_26)
4.
試問\(\displaystyle \lim_{n\to \infty}\left(\frac{sin\frac{\pi}{n}}{n}+\frac{sin\frac{2\pi}{n}}{n}+\frac{sin\frac{3\pi}{n}}{n}+\ldots+\frac{sin\frac{n\pi}{n}}{n}\right)\)之值為下列何者？
(A)0 (B)1 (C)2 (D)\(\displaystyle \frac{1}{\pi}\) (E)\(\displaystyle \frac{2}{\pi}\)
(105桃園高中，weiye解題
https://math.pro/db/viewthread.php?tid=2489&page=4#pid16492)
5.
設\(n\)為正整數，則\(C_1^n+3C_2^n+3^2C_3^n+3^3C_4^n+\ldots+3^{n-1}C_n^n=\)？
(A)\(\displaystyle \frac{4^n-1}{3}\) (B)\(\displaystyle \frac{4^n}{3}\) (C)\(\displaystyle \frac{4^n+1}{3}\) (D)\(4^n-1\) (E)\(4^n\)
(112新竹市國中聯招，
https://math.pro/db/thread-3763-1-1.html)
8.
已知實數\(x,y\)滿足條件\(\displaystyle sinx+siny=\frac{\sqrt{2}}{2}\)與\(\displaystyle cosx+cosy=\frac{\sqrt{6}}{2}\)，則\(sin(x+y)\)之值為何？
(A)\(\displaystyle \frac{\sqrt{2}}{4}\) (B)\(\displaystyle \frac{\sqrt{3}}{4}\) (C)1 (D)\(\displaystyle \frac{\sqrt{2}}{2}\) (E)\(\displaystyle \frac{\sqrt{3}}{2}\)
令\(a\)與\(b\)皆為實數且滿足\(\displaystyle sin a+sin b=\frac{\sqrt{2}}{2}\)，\(\displaystyle cos a+cos b=\frac{\sqrt{6}}{2}\)，試求出\(sin(a+b)\)之值。
(96中山大學雙週一題第1題，連結有三種解法
http://www.math.nsysu.edu.tw/~problem/2008s/962Q&A.htm)
15.
\(\Delta ABC\)中，\(\displaystyle tan\angle BAC=\frac{22}{7}\)，過頂點\(A\)作\(\overline{BC}\)邊上的高交\(\overline{BC}\)於\(D\)點，使得\(\overline{BD}=3,\overline{DC}=17\)，則\(\Delta ABC\)的面積為何？
(A)110 (B)120 (C)170 (D)220 (E)510
In triangle \(ABC\), \(\displaystyle \tan \angle CAB = \frac{22}{7}\), and the altitude from \(A\) divides \(BC\) into segments of length 3 and 17. What is the area of triangle \(ABC\)?
(1988AIME，
https://artofproblemsolving.com/ ... _Problems/Problem_7)
2.(D)
\(\displaystyle \sum_{n=1}^{\infty}\frac{1}{n\sqrt{n+1}+(n+1)\sqrt{n}}=\frac{1}{2}\)
(我的教甄準備之路 裂項相消，
https://math.pro/db/viewthread.php?tid=661&page=2#pid1678)
[提示]
\(\displaystyle \frac{1}{(n+1)\sqrt{n}+n\sqrt{n+1}}\times \frac{(n+1)\sqrt{n}-n\sqrt{n+1}}{(n+1)\sqrt{n}-n\sqrt{n+1}}=\frac{(n+1)\sqrt{n}-n\sqrt{n+1}}{n(n+1)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
