Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.

During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.

$p(x) $ is a real polynomial of degree $3$. Find necessary and sufficient conditions on its coefficients in order that $p(n)$ is integral for every integer $n$.

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) \equal{} S(Q(x)) \quad \forall x.$

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]

Let $f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)$. Show that at the point $x=1$, $f(x)$ and all its derivatives are irrational.

Prove that there exist positive constants $c_1$ and $c_2$ with the following properties: a) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$

Let $a$ be the unique real number $x$ satisfying $xe^x = 2$. Find a closed-form expression for \[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \] You may express your answer in terms of elementary operations, functions, and constants.

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$. Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin. (1) Find the equation of $l$. (2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

\[\int_2^0 x^2+3\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]