Olympiad problems

2009 Today's Calculation Of Integral, 471

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

2012 Today's Calculation Of Integral, 796

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

Answer the following questions: (1) Let $a$ be non-zero constant. Find $\int x^2 \cos (a\ln x)dx.$ (2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve $y=x\cos (\ln x)$, the $x$-axis and the lines $x=1,\ x=e^{\frac{\pi}{4}}$ about the $x$-axis.

2005 Today's Calculation Of Integral, 37

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

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

2023 CMIMC Integration Bee, 11

Tags: integration , calculus

\[\int_{-1}^1 \frac{1}{(8 + x^2)\sqrt{1-x^2}} \,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Unirea, 4

Tags: function , parameterization , inequalities , integration , real analysis

$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable suppose $f(a)=f(-a)$ $ f'(-a)=f'(a)=a^2$ Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$ Study equality case ? Radu Miculescu

2010 SEEMOUS, Problem 1

Tags: integration , function , sequence , convergence , calculus

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$. a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$. b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

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$.

2008 AIME Problems, 13

Tags: geometry , reflection , geometric transformation , integration , analytic geometry , trigonometry , calculus

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.

2006 Mathematics for Its Sake, 2

Tags: integration , calculus

Calculate: [b]a)[/b] $ \int \frac{1-x^2-x^6+x^8}{1+x^{10}} dx $ [b]b)[/b] $ \int\frac{x^{n-1}+x^{5n-1}}{1+x^{6n}} dx $ [i]Dumitru Acu[/i]

2007 Estonia National Olympiad, 3

Tags: analytic geometry , integration , calculus , geometry

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

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 .$

2016 Danube Mathematical Olympiad, 3

Tags: integration , inequalities , calculus

3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that $\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$, where $a_0 = 0$. b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.

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. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

2009 Jozsef Wildt International Math Competition, W. 14

Tags: integration , inequalities , calculus

If the function $f:[0,1]\to (0.+\infty)$ is increasing and continuous, then for every $a\geq 0$ the following inequality holds: $$\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx$$

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]

2010 IMC, 2

Tags: integration , infinite series , logarithm , real analysis

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

2005 Harvard-MIT Mathematics Tournament, 3

Tags: integration , function , calculus

Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?

2014 Putnam, 2

Tags: function , absolute value , real analysis , integration

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

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).