Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $? $ \textbf{(A)} \ (\log 2016, \log 2017) $ $ \textbf{(B)} \ (\log 2017, \log 2018) $ $ \textbf{(C)} \ (\log 2018, \log 2019) $ $ \textbf{(D)} \ (\log 2019, \log 2020) $ $ \textbf{(E)} \ (\log 2020, \log 2021) $

Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that $$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$

Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

Two different functions $f, g$ of $x$ are selected from the set of real-valued functions $$\left \{sin x, e^{-x}, x \ln x, \arctan x, \sqrt{x^2 + x} -\sqrt{x^2 + x} -x, \frac{1}{x} \right \}$$ to create a product function $f(x)g(x)$. For how many such products is $\lim_{x\to infty} f(x)g(x)$ finite?

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that $\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$ $\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.

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

For $a,b,c>0: \; abc=1$ prove that \[a^3+b^3+c^3+6 \ge (a+b+c)^2\]

Given any set of $9$ points in the plane such that there is no $3$ of them collinear, show that for each point $P$ of the set, the number of triangles with its vertices on the other $8$ points and that contain $P$ on its interior is even.

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.