The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Calculate the following indefinite integrals. [1] $\int \sin (\ln x)dx$ [2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$ [3] $\int \frac{x^3}{x^2+1}dx$ [4] $\int \frac{x^2}{\sqrt{2x-1}}dx$ [5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that \[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\] Prove that \[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\] [i]Cezar Lupu[/i]

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.

Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that \[ \frac{\log_{10} m}{\log_{10} n}\] is not a rational number.

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that \[a_{m^2+n^2}=a_m^2+a_n^2 \] for all integers $m,n\ge 0$.

Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel has $k$ sheets of paper lying next to each other on a table, where $k$ is a positive integer. On each of the sheets, he writes some of the numbers from $1$ up to $n$ (he is allowed to write no number at all, or all numbers). On the back of each of the sheets, he writes down the remaining numbers. Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making all of the numbers from $1$ up to n visible at least once, then he wins. Determine the smallest $k$ for which Merlijn can always win, regardless of Daniel’s actions.

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that \[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}} \]is defined? $ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

Solve \[ \left\{ \begin{array}{l} \log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\ \log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\ \log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\ \end{array} \right.\]

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

If $x > 10$, what is the greatest possible value of the expression \[ {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ? \] All the logarithms are base 10.

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.