Olympiad problems

2009 Today's Calculation Of Integral, 426

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

2011 Today's Calculation Of Integral, 722

Tags: calculus , integration , function , calculus computations

Find the continuous function $f(x)$ such that : \[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]

2007 Today's Calculation Of Integral, 229

Tags: calculus , integration , limit , trigonometry , function , calculus computations

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

2005 Today's Calculation Of Integral, 32

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

2010 Today's Calculation Of Integral, 531

Tags: calculus , integration , function , trigonometry , geometry , rectangle , calculus computations

(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that \[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\] Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$, Prove that there exists $ \theta$ such that \[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]

2012 Today's Calculation Of Integral, 794

Tags: calculus , integration , function , trigonometry , calculus computations

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

2005 Today's Calculation Of Integral, 35

Tags: calculus , integration , derivative , function , calculus computations

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

2011 Today's Calculation Of Integral, 710

Tags: calculus , integration , trigonometry , calculus computations , logarithm

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

2012 Today's Calculation Of Integral, 834

Tags: calculus , integration , geometry , calculus computations

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

2010 Today's Calculation Of Integral, 574

Tags: calculus , integration , calculus computations , logarithm

Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2010 Today's Calculation Of Integral, 647

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

2011 Today's Calculation Of Integral, 680

Tags: calculus , integration , calculus computations

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2010 Today's Calculation Of Integral, 547

Tags: calculus , integration , function , derivative , real analysis , calculus computations , logarithm

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2013 Today's Calculation Of Integral, 863

Tags: calculus , integration , trigonometry , limit , derivative , inequalities , calculus computations

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2009 Today's Calculation Of Integral, 480

Tags: calculus , integration , inequalities , calculus computations , geometry

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

2010 Today's Calculation Of Integral, 620

Tags: calculus , integration , function , trigonometry , parabola , calculus computations , conic

Let $a,\ b$ be real numbers. Suppose that a function $f(x)$ satisfies $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$ and has the maximum value $2\pi$ for $-\pi \leq x\leq \pi$. Find the minimum value of $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$ [i]2010 Chiba University entrance exam[/i]

Today's calculation of integrals, 877

Tags: calculus , integration , trigonometry , calculus computations , definite integral , limit

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2007 Today's Calculation Of Integral, 170

Tags: calculus , integration , algebra , partial fractions , calculus computations , logarithm

Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2009 Today's Calculation Of Integral, 436

Tags: calculus , integration , geometry , calculus computations

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

2010 Today's Calculation Of Integral, 619

Tags: calculus , integration , function , trigonometry , calculus computations , logarithm

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

2007 Today's Calculation Of Integral, 183

Tags: calculus , integration , limit , trigonometry , calculus computations

Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows. [1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$ [2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

Today's calculation of integrals, 886

Tags: calculus , integration , function , trigonometry , calculus computations

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2005 Today's Calculation Of Integral, 88

Tags: calculus , integration , function , calculus computations

A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$ Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$

2011 Today's Calculation Of Integral, 702

Tags: calculus , integration , function , calculus computations

$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$

2006 ISI B.Stat Entrance Exam, 6

Tags: function , limit , calculus , calculus computations

(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$. (b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.