Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

For every positive integer $n$, let \[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\] Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.

Define $f(n):$ the number of integral points of line segment $OA_n$ ($O$ and $A_n$ not included), where $A_n(n,n+3)$. Then, $f(1)+f(2)+\cdots+f(1990)=$________.

Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.

For a function $ f(x) \equal{} 6x(1 \minus{} x)$, suppose that positive constant $ c$ and a linear function $ g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0)$ satisfy the following 3 conditions: $ c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0$. Answer the following questions. (1) Find the constants $ a,\ b,\ c$. (2) For natural number $ n$, let $ I_n \equal{} \int_0^1 x^ne^x\ dx$. Express $ I_{n \plus{} 1}$ in terms of $ I_n$. Then evaluate $ I_1,\ I_2,\ I_3$. (3) Evaluate the definite integrals $ \int_0^1 e^xf(x)\ dx$ and $ \int_0^1 e^xf(x)g(x)\ dx$. (4) For real numbers $ s,\ t$, define $ J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx$. Find the constants $ A,\ B,\ C,\ D,\ E$ by setting $ J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F$. (You don't need to find the constant $ F$). (5) Find the values of $ s,\ t$ for which $ J$ is minimal.

Suppose that $ f: [a,b]\rightarrow \mathbb{R} $ be a monotonic function and for every $ x_1,x_2\in [a,b] $ that $ x_1<x_2 $ ,there exist $ c\in (a,b) $ such that $ \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) $ a) Show that $ f $ be the continuous function on interval $ (a,b) $ b) Suppose that $ f $ is integrable function on interval $ [a,b] $ but $ f $ isn't a monotonic function then ,is it the result of part a) right?

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

Let $p>1,\frac1p+\frac1q=1$ and $r>1$. If $u(x,y),v(x,y)>0$, and $f(x,y),g(x,y)$ are continuous functions on $[a,b]\times[c,d]$, then prove $$\left(\frac{\left(\int^b_a\int^d_c(f(x,y)+g(x,y))^rdxdy\right)^{1/r}}{(u(x,y)+v(x,y))^{1/q}}\right)^p\le\left(\frac{\left(\int^b_a\int^d_cf(x,y)^rdxdy\right)^{1/r}}{u(x,y)^{1/q}}\right)^p+\left(\frac{\left(\int^b_a\int^d_cg(x,y)^rdxdy\right)^{1/r}}{v(x,y)^{1/q}}\right)^p,$$ with equality if and only if either $$\left(\lVert f(x,y)\rVert^r_r,\lVert g(x,y)\rVert^r_r\right)=\alpha\left(\lVert u(x,y)\rVert^r_r,\lVert v(x,y)\rVert^r_r\right)$$ for some $\alpha>0$ or $\lVert f(x,y)\rVert^r_r=\lVert g(x,y)\rVert^r_r=0$. [i]Proposed by Chang-Jian Zhao[/i]

For a real number $ x$ in the interval $ (0,1)$ with decimal representation \[ 0.a_1(x)a_2(x)...a_n(x)...,\] denote by $ n(x)$ the smallest nonnegative integer such that \[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\] Determine $ \int_0^1n(x)dx$. ($ \overline{abcd}$ denotes the decimal number with digits $ a,b,c,d .$) [i]A. Renyi[/i]

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

Problem 3. Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that: (a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

Find all differentiable functions $ f:(0,\infty )\longrightarrow (-\infty ,\infty ) $ having the property that $$ f'(\sqrt{x}) =\frac{1+x+x^2}{1+x} , $$ for any positive real numbers $ x. $ [i]Ovidiu Țâțan[/i]

Let $ a$ be a positive real number, in Euclidean space, consider the two disks: $ D_1\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}a\}$, $ D_2\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}\minus{}a\}$. Let $ D_1$ overlap to $ D_2$ by rotating $ D_1$ about the $ y$ axis by $ 180^\circ$. Note that the rotational direction is supposed to be the direction such that we would lean the postive part of the $ z$ axis to into the direction of the postive part of $ x$ axis. Let denote $ E$ the part in which $ D_1$ passes while the rotation, let denote $ V(a)$ the volume of $ E$ and let $ W(a)$ be the volume of common part of $ E$ and $ \{(x,\ y,\ z)|x\geq 0\}$. (1) Find $ W(a)$. (2) Find $ \lim_{a\rightarrow \infty} V(a)$.

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

Find integral: $\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous function such that $$ \int_0^1 (x-1)f(x)dx =0. $$ Show that: [b]a)[/b] There exists $ a\in (0,1) $ such that $ \int_0^a xf(x)dx =0. $ [b]b)[/b] There exists $ b\in (0,1) $ so that $ \int_0^b xf(x)dx=bf(b). $

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.

Evaluate \[ \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]

For positive integers $ m$ and $ n$, let $ f\left(m,n\right)$ denote the number of $ n$-tuples $ \left(x_1,x_2,\dots,x_n\right)$ of integers such that $ \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m$. Show that $ f\left(m,n\right) \equal{} f\left(n,m\right)$.

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

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