Olympiad problems

2008 ISI B.Stat Entrance Exam, 3

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

2011 Today's Calculation Of Integral, 674

Tags: integration , calculus , logarithm , calculus computations

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]

2007 Today's Calculation Of Integral, 172

Tags: trigonometry , integration , calculus computations , euler , calculus

Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

2013 Today's Calculation Of Integral, 894

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

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

2008 ISI B.Stat Entrance Exam, 6

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

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2010 Today's Calculation Of Integral, 650

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

Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds: \[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\] [i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]

2007 Today's Calculation Of Integral, 183

Tags: limit , integration , calculus , 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 $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

2005 Today's Calculation Of Integral, 84

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

Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]

2009 Today's Calculation Of Integral, 432

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

Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.

2005 Today's Calculation Of Integral, 43

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

Evaluate \[\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx\]

2012 Today's Calculation Of Integral, 803

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

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2013 District Olympiad, 1

Tags: limit , calculus , inequalities , calculus computations

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

2007 Today's Calculation Of Integral, 204

Tags: trigonometry , integration , special factorizations , calculus , calculus computations

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

2007 Today's Calculation Of Integral, 210

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

Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.

2009 Today's Calculation Of Integral, 491

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

Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ (1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$. (2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$

2005 Today's Calculation Of Integral, 12

Tags: calculus , integration , trigonometry , calculus computations

Calculate the following indefinite integrals. [1] $\int \frac{dx}{1+\cos x}$ [2] $\int x\sqrt{x^2-1}dx$ [3] $\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)$ [4] $\int \frac{\sin ^ 3 x}{1+\cos x}dx$ [5] $\int e^{4x}\sin 2x dx$

2010 Contests, 523

Tags: logarithm , integration , calculus computations , calculus

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2010 ISI B.Math Entrance Exam, 5

Tags: calculus , limit , calculus computations

Let $a_1>a_2>.....>a_r$ be positive real numbers . Compute $\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}$

2010 Today's Calculation Of Integral, 625

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

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2010 Today's Calculation Of Integral, 599

Tags: trigonometry , integration , calculus , calculus computations

Evaluate $\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx$. created by kunny

2010 Today's Calculation Of Integral, 570

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

Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$. (1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$. (2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$. (3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.