Olympiad problems

2008 Harvard-MIT Mathematics Tournament, 9

Tags: limit , integration , hmmt , calculus , real analysis , calculus computations , logarithm

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2009 Today's Calculation Of Integral, 400

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

(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

2007 Today's Calculation Of Integral, 233

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

Find the minimum value of the following definite integral. $ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$

2005 Gheorghe Vranceanu, 3

Tags: limit , euler s number , real analysis , calculus

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

2012 Today's Calculation Of Integral, 840

Tags: calculus , integration , function , trigonometry , algebra , domain , inequalities

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 5

Tags: calculus , integration , limit , complex analysis , real analysis

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

2009 Today's Calculation Of Integral, 464

Tags: calculus , integration , calculus computations , logarithm

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

1989 China Team Selection Test, 3

Tags: gauss , calculus , derivative , algebra , polynomial , combinatorial geometry

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

2009 Miklós Schweitzer, 9

Tags: calculus , integration , inequalities , function , invariant , real analysis

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

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 $f(x)=\frac{c}{x}\ (c: constant).$

2007 Today's Calculation Of Integral, 214

Tags: calculus , integration , geometry , calculus computations

Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.

2004 USAMO, 3

Tags: geometry , ratio , function , calculus , quadratic

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

1995 IMC, 2

Tags: function , calculus , real analysis , integral , inequalities

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

1974 AMC 12/AHSME, 10

Tags: calculus , integration , quadratic

What is the smallest integral value of $k$ such that \[ 2x(kx-4)-x^2+6=0 \] has no real roots? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2024 CMIMC Integration Bee, 8

Tags: calculus , integration

\[\int_1^2 \cos(\sin^{-1}(\tan(\cos^{-1}(\sin(\tan^{-1}(x))))))\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2011 Today's Calculation Of Integral, 683

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$. [i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]

2005 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , trigonometry

Compute \[ \displaystyle\sum_{k=0}^{\infty} \dfrac {4}{(4k)!}. \]

2003 District Olympiad, 2

Tags: function , integral , calculus , integration , ratio , sequence

Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as $$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$ [b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $ [b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.

2011 AMC 10, 13

Tags: probability , ratio , 3d geometry , tetrahedron , limit , calculus , geometry

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3} $