Olympiad problems

2006 Harvard-MIT Mathematics Tournament, 10

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

2011 Today's Calculation Of Integral, 735

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

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2009 Today's Calculation Of Integral, 498

Tags: calculus computations , integration , function , calculus

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

Today's calculation of integrals, 859

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

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2022 VJIMC, 3

Tags: calculus , function , integration

Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit $$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$

2010 Morocco TST, 2

Tags: algebra , series summation , approximation , floor function , integration , calculus

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

2007 Today's Calculation Of Integral, 180

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

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2009 Today's Calculation Of Integral, 480

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

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

1957 Putnam, B3

Tags: inequalities , function , integration

For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that $$ \int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$

2010 Today's Calculation Of Integral, 571

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

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

2023 CMIMC Integration Bee, 10

Tags: integration , calculus

\[\int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac{\arctan(x)\log^2(x)}{x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Harvard-MIT Mathematics Tournament, 10

Tags: function , trigonometry , derivative , integration , calculus , geometric series

Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.

2011 VTRMC, Problem 1

Tags: calculus , integration

Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$

2016 ISI Entrance Examination, 7

Tags: real analysis , integration , calculus

$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of $$\int_0^1(x-f(x))^{2016}dx$$

2010 Today's Calculation Of Integral, 613

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

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

2023 CMIMC Integration Bee, 3

Tags: calculus , integration

\[\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2014 SEEMOUS, Problem 4

Tags: calculus , limit , integration

a) Prove that $\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2$. b) Find the limit $\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right)$.

1995 AIME Problems, 11

Tags: rectangle , integration , 3d geometry , ratio , prism , calculus , geometry

A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

2010 District Olympiad, 4

Tags: real analysis , integration , inequalities , function

Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$. i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing. ii)Prove that for each integer number $ n\ge 1$, we have: $ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$

2011 Today's Calculation Of Integral, 682

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

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

2013 Today's Calculation Of Integral, 898

Tags: integration , logarithm , calculus , calculus computations

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2019 Jozsef Wildt International Math Competition, W. 16

Tags: inequalities , calculus , integration , trigonometry

If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$

2010 Today's Calculation Of Integral, 636

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

Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$ (1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$. (2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$. [i]1992 Tokyo University entrance exam/Science[/i]

2013 Bogdan Stan, 2

Tags: function , integration , absolute value , calculus

Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties: $ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $ $ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $ Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $ [i]Cristinel Mortici[/i]

2014 Contests, 901

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

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$