Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$

Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

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

Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds: \[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\] [i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]

Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}$.

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

Evaluate the following definite integral. \[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

If $ a,b,c\in {\rm Z}$ and \[ \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\ {x\equiv b\, \, \, \pmod {15}} \\ {x\equiv c\, \, \, \pmod {16}} \end{array} \] , the number of integral solutions of the congruence system on the interval $ 0\le x < 2000$ cannot be $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

Find the minimum value of\[ (u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2 \]for $0<u<\sqrt{2}$ and $v>0$

Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$