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2.

(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)$$\displaystyle \frac{1+\sqrt{5}}{2}$$

(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) $$\displaystyle ( 3,\frac{7}{2} )$$

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.

(A)0　(B)1　(C)2　(D)$$\displaystyle \frac{1}{\pi}$$　(E)$$\displaystyle \frac{2}{\pi}$$

5.

(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)$$4^n$$

8.

(A)$$\displaystyle \frac{\sqrt{2}}{4}$$　(B)$$\displaystyle \frac{\sqrt{3}}{4}$$　(C)1　(D)$$\displaystyle \frac{\sqrt{2}}{2}$$　(E)$$\displaystyle \frac{\sqrt{3}}{2}$$

(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)
$$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n\sqrt{n+1}+(n+1)\sqrt{n}}=\frac{1}{2}$$
[提示]
$$\displaystyle \frac{1}{(n+1)\sqrt{n}+n\sqrt{n+1}}\times \frac{(n+1)\sqrt{n}-n\sqrt{n+1}}{(n+1)\sqrt{n}-n\sqrt{n+1}}=\frac{(n+1)\sqrt{n}-n\sqrt{n+1}}{n(n+1)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}$$

2007 TRML 團體賽

http://www.shiner.idv.tw/teachers/viewtopic.php?t=2551

[ 本帖最後由 thepiano 於 2021-8-11 22:24 編輯 ]

A可對角化時表示存在Q (可逆) such that Q^(-1)AQ=D

7.

[ 本帖最後由 satsuki931000 於 2022-1-9 11:05 編輯 ]

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