Olympiad problems

2010 Today's Calculation Of Integral, 644

For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

2011 Today's Calculation Of Integral, 753

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

Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

2010 Today's Calculation Of Integral, 547

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

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

2011 Today's Calculation Of Integral, 676

Tags: calculus computations , calculus , integration , trigonometry

Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$. Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$. [i]2011 Tokyo University of Science entrance exam/Management[/i]

Today's calculation of integrals, 850

Tags: calculus computations , trigonometry , calculus , integration

Evaluate \[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]

2007 Today's Calculation Of Integral, 240

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

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

2010 Today's Calculation Of Integral, 577

Tags: calculus , integration , calculus computations , inequalities

Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

2010 Today's Calculation Of Integral, 594

Tags: calculus computations , integration , geometry , calculus

In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$. Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.

Today's calculation of integrals, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2009 Today's Calculation Of Integral, 401

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

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

2008 Harvard-MIT Mathematics Tournament, 5

Tags: derivative , calculus , trigonometry , calculus computations

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

2007 Today's Calculation Of Integral, 229

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

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

2010 Today's Calculation Of Integral, 645

Tags: integration , calculus computations , calculus , inequalities , function

Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

2013 District Olympiad, 3

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

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 $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

2012 Today's Calculation Of Integral, 800

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

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

2014 Contests, 2

Tags: parabola , calculus , integration , geometry , conic , calculus computations

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2005 Today's Calculation Of Integral, 82

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

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

2013 Today's Calculation Of Integral, 881

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

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2010 Today's Calculation Of Integral, 652

Tags: calculus computations , calculus , integration

Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

2012 Today's Calculation Of Integral, 773

Tags: calculus computations , calculus , logarithm , integration

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

2009 Today's Calculation Of Integral, 493

Tags: calculus computations , calculus , trigonometry , conic , ellipse , geometry , integration

In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.

2007 Today's Calculation Of Integral, 249

Tags: logarithm , calculus , integration , calculus computations

Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2012 Today's Calculation Of Integral, 770

Tags: calculus computations , calculus , integration

Find the value of $a$ such that : \[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]

2009 Today's Calculation Of Integral, 464

Tags: logarithm , calculus , integration , calculus computations

Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.

Today's calculation of integrals, 858

Tags: perimeter , calculus computations , calculus , integration , geometry

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 $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.