Olympiad problems

2012 Today's Calculation Of Integral, 813

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

2008 Harvard-MIT Mathematics Tournament, 7

Tags: 3d geometry , derivative , limit , geometry , calculus computations , calculus , function

([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

2011 Today's Calculation Of Integral, 765

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

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.

2010 Today's Calculation Of Integral, 651

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

Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

2010 Today's Calculation Of Integral, 629

Tags: calculus , integration , calculus computations

Evaluate $\int_0^{\infty} \frac{1}{e^{x}(1+e^{4x})}dx.$

2005 Today's Calculation Of Integral, 27

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

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

2011 Today's Calculation Of Integral, 749

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

Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.

2009 Today's Calculation Of Integral, 403

Tags: calculus , integration , calculus computations

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

1978 Putnam, A3

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

Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.

2010 District Olympiad, 4

Tags: calculus , limit , calculus computations

Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that: \[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]

2010 Today's Calculation Of Integral, 541

Tags: function , integration , calculus computations , calculus

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)$

2012 Today's Calculation Of Integral, 849

Tags: calculus , integration , calculus computations , function

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

2009 Today's Calculation Of Integral, 476

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

Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.

2012 Today's Calculation Of Integral, 816

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

Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

2012 Today's Calculation Of Integral, 828

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

Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$

Today's calculation of integrals, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2011 Today's Calculation Of Integral, 693

Tags: trigonometry , calculus , calculus computations , integration

Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$ created by kunny

2009 Today's Calculation Of Integral, 474

Tags: integration , calculus computations , trigonometry , calculus

Calculate the following indefinite integrals. (1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$ (2) $ \int \sin 2x\cos 2x\cos 4x\ dx$ (3) $ \int xe^{x}dx$ (4) $ \int 5^{x}dx$

Today's calculation of integrals, 883

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

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

2007 Today's Calculation Of Integral, 213

Tags: calculus , calculus computations , integration

Find the minimum value of $ f(a)=\int_{0}^{1}x|x-a|\ dx$.

2008 Harvard-MIT Mathematics Tournament, 1

Tags: polynomial , algebra , calculus , function , calculus computations

Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

2010 Today's Calculation Of Integral, 597

Tags: calculus computations , integration , calculus

In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes. 1984 Tokyo University entrance exam/Science

Today's calculation of integrals, 871

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

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2009 Today's Calculation Of Integral, 499

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

Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

2009 Today's Calculation Of Integral, 520

Tags: calculus , calculus computations , integration

Let $ a,\ b,\ c$ be postive constants. Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.