Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

If $ a$ and $ b$ are positive numbers such that $ a^b \equal{} b^a$ and $ b \equal{} 9a$, then the value of $ a$ is: $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \sqrt [9] {9}\qquad \textbf{(D)}\ \sqrt [3] {9}\qquad \textbf{(E)}\ \sqrt [4] {3}$

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

$(a)$ Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*} $(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\ \\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\ \\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers $x^y$, $y^x$ is bigger ?

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$? $ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 154 \qquad \textbf{(D)}\ 406 \qquad \textbf{(E)}\ 1024$

Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]

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]

Show that there exist two consecutive squares such that there are at least $1000$ primes between them.

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

Let $0<a<b$.Prove the following inequaliy. \[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]

If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$ $\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).

Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.