$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

\[\lim_{n\to\infty} n^2 \int_0^1 x^n e^{-x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon and Vlad Oleksenko[/i]

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$.

As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all. Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?" "Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines. Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.

Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

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.

Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \dfrac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \dfrac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\dfrac{\text{ft}}{\text{s}}$)?

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$. [i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]