Olympiad problems

2009 Today's Calculation Of Integral, 442

Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

2007 Today's Calculation Of Integral, 173

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

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

2010 Today's Calculation Of Integral, 647

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

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

Today's calculation of integrals, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2013 Today's Calculation Of Integral, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2009 Today's Calculation Of Integral, 484

Tags: calculus computations , integration , geometry , calculus , trapezoid , limit , logarithm

Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

2010 Today's Calculation Of Integral, 632

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

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

2009 Today's Calculation Of Integral, 460

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

$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.

2013 Today's Calculation Of Integral, 887

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

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.

2008 Harvard-MIT Mathematics Tournament, 4

Tags: quadratic , limit , logarithm , calculus , calculus computations

([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.

2012 Today's Calculation Of Integral, 783

Tags: calculus computations , integration , calculus

Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited

2009 Today's Calculation Of Integral, 471

Tags: calculus computations , logarithm , calculus , integration

Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

2005 Today's Calculation Of Integral, 37

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

Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

2005 Today's Calculation Of Integral, 46

Tags: integration , logarithm , calculus , calculus computations

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

2009 Today's Calculation Of Integral, 428

Tags: calculus , integration , polynomial , algebra , calculus computations

Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

2007 Today's Calculation Of Integral, 246

Tags: polynomial , geometry , integration , algebra , calculus , function , calculus computations

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

2005 Today's Calculation Of Integral, 57

Tags: calculus , calculus computations , trigonometry , integration

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

2007 Today's Calculation Of Integral, 228

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

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

2010 Today's Calculation Of Integral, 658

Tags: trigonometry , integration , analytic geometry , calculus computations , calculus , geometry

Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

2007 Today's Calculation Of Integral, 254

Tags: logarithm , integration , calculus , calculus computations

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

2010 Today's Calculation Of Integral, 559

Tags: geometric transformation , rotation , trigonometry , integration , calculus , calculus computations , geometry

In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions. (1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$. (2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$. (3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$. Then use this to find the area of (2).