For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$

Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$. [i]Marius Cavachi[/i]

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 77 $

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

Let $d(n)$ be the number of positive integers that divide the integer $n$. For all positive integral divisors $k$ of $64800$, what is the sum of numbers $d(k)$? $ \textbf{(A)}\ 1440 \qquad\textbf{(B)}\ 1650 \qquad\textbf{(C)}\ 1890 \qquad\textbf{(D)}\ 2010 \qquad\textbf{(E)}\ \text{None of above} $

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$ Sorry, I have deleted my first post because that was wrong. kunny

Find all integers $x$ such that $2x^2+x-6$ is a positive integral power of a prime positive integer.

Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

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) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

Evaluate \[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]

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

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.