Olympiad problems

2009 Today's Calculation Of Integral, 494

Tags: calculus , integration , hyperbola , analytic geometry , calculus computations , conic

Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

2013 Today's Calculation Of Integral, 866

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2010 Today's Calculation Of Integral, 559

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

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).

2010 Today's Calculation Of Integral, 538

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

2009 Today's Calculation Of Integral, 472

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

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

2005 Today's Calculation Of Integral, 85

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

Evaluate \[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\] where $ [x] $ is the integer equal to $ x $ or less than $ x $.

2011 Today's Calculation Of Integral, 713

Tags: calculus , integration , limit , calculus computations

If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$

2009 Today's Calculation Of Integral, 502

Tags: calculus , integration , limit , calculus computations

(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$ (2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.

2009 Today's Calculation Of Integral, 416

Tags: calculus , integration , trigonometry , calculus computations

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

2010 Today's Calculation Of Integral, 523

Tags: calculus , integration , calculus computations , logarithm

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

2011 Today's Calculation Of Integral, 745

Tags: calculus , integration , trigonometry , calculus computations

When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$

2009 Today's Calculation Of Integral, 445

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2012 Today's Calculation Of Integral, 771

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

(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

2009 Today's Calculation Of Integral, 408

Tags: calculus , integration , calculus computations , logarithm

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

2010 Today's Calculation Of Integral, 585

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

2007 Moldova National Olympiad, 11.2

Tags: limit , calculus , logarithm , calculus computations

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

2005 Today's Calculation Of Integral, 64

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

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

2007 Today's Calculation Of Integral, 185

Tags: calculus , integration , trigonometry , calculus computations

Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

Today's calculation of integrals, 894

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

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

2011 Today's Calculation Of Integral, 715

Tags: calculus , integration , function , calculus computations

Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3

Tags: function , limit , calculus , calculus computations

Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.

2012 Today's Calculation Of Integral, 813

Tags: calculus , integration , geometry , inequalities , quadratic , calculus computations

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

2013 Today's Calculation Of Integral, 883

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

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

Today's calculation of integrals, 899

Tags: calculus , integration , limit , calculus computations

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2010 Today's Calculation Of Integral, 541

Tags: calculus , integration , function , calculus computations

Find the functions $ f(x),\ g(x)$ satisfying the following equations. (1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$ (2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$