Olympiad problems

2008 Harvard-MIT Mathematics Tournament, 2

([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

2009 Today's Calculation Of Integral, 487

Tags: function , integration , calculus , calculus computations

Suppose two functions $ f(x)\equal{}x^4\minus{}x,\ g(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ satisfy $ f(1)\equal{}g(1),\ f(\minus{}1)\equal{}g(\minus{}1)$. Find the values of $ a,\ b,\ c,\ d$ such that $ \int_{\minus{}1}^1 (f(x)\minus{}g(x))^2dx$ is minimal.

2007 Today's Calculation Of Integral, 226

Tags: calculus , calculus computations , trigonometry , integration

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx$ [color=darkblue]Virgil Nicula have already posted the integral[/color] :oops:

2010 Today's Calculation Of Integral, 549

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

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

2012 Today's Calculation Of Integral, 846

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

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

2007 Today's Calculation Of Integral, 233

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

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

2009 Today's Calculation Of Integral, 422

Tags: integration , calculus computations , calculus , probability

There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time. Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.

Today's calculation of integrals, 900

Tags: trigonometry , integration , calculus , calculus computations

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

2009 Today's Calculation Of Integral, 435

Tags: calculus , integration , calculus computations , trigonometry

Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.

2009 Today's Calculation Of Integral, 408

Tags: logarithm , calculus computations , integration , calculus

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

2007 Today's Calculation Of Integral, 227

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

Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$

2010 Today's Calculation Of Integral, 639

Tags: integration , calculus , calculus computations

Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

2010 Today's Calculation Of Integral, 620

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

Let $a,\ b$ be real numbers. Suppose that a function $f(x)$ satisfies $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$ and has the maximum value $2\pi$ for $-\pi \leq x\leq \pi$. Find the minimum value of $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$ [i]2010 Chiba University entrance exam[/i]

2011 Today's Calculation Of Integral, 740

Tags: calculus , geometry , derivative , integration , calculus computations

Let $r$ be a positive constant. If 2 curves $C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}$ have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by $C_1,\ C_2$.

2009 Today's Calculation Of Integral, 475

Tags: integration , calculus , calculus computations

For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$

2009 Today's Calculation Of Integral, 489

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

Find the following limit. $ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.

2009 Today's Calculation Of Integral, 451

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

Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).$

1962 Vietnam National Olympiad, 2

Tags: calculus , derivative , function , calculus computations

Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.

2005 Today's Calculation Of Integral, 30

Tags: calculus , integration , partial fractions , calculus computations , algebra

A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$ Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

2010 Today's Calculation Of Integral, 648

Tags: calculus , integration , calculus computations , function

Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that: (a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$ (b) For $x\neq 0,\ f(x)>0.$ Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer. $(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\] $(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\] [i]2010 Kyoto University, Master Course in Mathematics[/i]

2010 Today's Calculation Of Integral, 529

Tags: calculus computations , integration , trigonometry , inequalities , calculus

Prove that the following inequality holds for each natural number $ n$. \[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]

2012 Today's Calculation Of Integral, 858

Tags: calculus , geometry , perimeter , integration , calculus computations

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

2007 Today's Calculation Of Integral, 209

Tags: calculus , integration , calculus computations , trigonometry

Let $m,\ n$ be the given distinct positive integers. Answer the following questions. (1) Find the real number $\alpha \ (|\alpha |<1)$ such that $\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0$. (2) Find the real number $\beta$ satifying the sytem of equation $\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}$, $\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}$.

2010 Today's Calculation Of Integral, 586

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

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

2013 Today's Calculation Of Integral, 864

Tags: limit , calculus , integration , calculus computations , inequalities

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]