Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.

Finished with rereading Isaac Asimov's $\textit{Foundation}$ series, Joshua asks his father, "Do you think somebody will build small devices that run on nuclear energy while I'm alive?" "Honestly, Josh, I don't know. There are a lot of very different engineering problems involved in designing such devices. But technology moves forward at an amazing pace, so I won't tell you we can't get there in time for you to see it. I $\textit{did}$ go to a graduate school with a lady who now works on $\textit{portable}$ nuclear reactors. They're not small exactly, but they aren't nearly as large as most reactors. That might be the first step toward a nuclear-powered pocket-sized video game. Hannah adds, "There are already companies designing batteries that are nuclear in the sense that they release energy from uranium hydride through controlled exoenergetic processes. This process is not the same as the nuclear fission going on in today's reactors, but we can certainly call it $\textit{nuclear energy}$." "Cool!" Joshua's interest is piqued. Hannah continues, "Suppose that right now in the year $2008$ we can make one of these nuclear batteries in a battery shape that is $2$ meters $\textit{across}$. Let's say you need that size to be reduced to $2$ centimeters $\textit{across}$, in the same proportions, in order to use it to run your little video game machine. If every year we reduce the necessary volume of such a battery by $1/3$, in what year will the batteries first get small enough?" Joshua asks, "The battery shapes never change? Each year the new batteries are similar in shape - in all dimensions - to the bateries from previous years?" "That's correct," confirms Joshua's mother. "Also, the base $10$ logarithm of $5$ is about $0.69897$ and the base $10$ logarithm of $3$ is around $0.47712$." This makes Joshua blink. He's not sure he knows how to use logarithms, but he does think he can compute the answer. He correctly notes that after $13$ years, the batteries will already be barely more than a sixth of their original width. Assuming Hannah's prediction of volume reduction is correct and effects are compounded continuously, compute the first year that the nuclear batteries get small enough for pocket video game machines. Assume also that the year $2008$ is $7/10$ complete.

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$. \[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\] If necessary, you may use $ \ln 3 \equal{} 1.10$.

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: ${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $

Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

Evaluate: \[\frac{1}{\log_2 (\frac{1}{6})} - \frac{1}{\log_3 (\frac{1}{6})} - \frac{1}{\log_4 (\frac{1}{6})}\]

Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor .$ [i]Petre Rău[/i]

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

Is $\log_{10}{8}$ rational?

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 $c=z.$

Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$