Olympiad problems

2022 VJIMC, 3

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

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$

2018 Romania National Olympiad, 2

Tags: function , calculus , integration , real analysis

Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

2007 Harvard-MIT Mathematics Tournament, 15

Tags: calculus , derivative

Points $A$, $B$, and $C$ lie in that order on line $\ell$ such that $AB=3$ and $BC=2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.

2008 Brazil National Olympiad, 3

Tags: calculus , trigonometry , inequalities

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

1989 IMO Longlists, 27

Tags: calculus , integration , modular arithmetic , number theory , relatively prime , combinatorics

Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

2007 Today's Calculation Of Integral, 248

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

Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

Today's calculation of integrals, 887

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

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

1963 Putnam, A3

Tags: calculus , integration , function , differential equation

Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$

2007 Today's Calculation Of Integral, 198

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

Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2020 LIMIT Category 2, 16

Tags: limit , derivative , calculus

The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$. Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

2014 Contests, 2

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

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2010 Today's Calculation Of Integral, 667

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

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

2005 Today's Calculation Of Integral, 53

Tags: calculus , integration , trigonometry , calculus computations

Find the maximum value of the following integral. \[\int_0^{\infty} e^{-x}\sin tx\ dx\]

2010 Today's Calculation Of Integral, 659

Tags: calculus , integration , geometric series , calculus computations , logarithm

Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$

2004 IMO Shortlist, 6

Tags: function , calculus , algebra , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

2010 Today's Calculation Of Integral, 595

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

Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

2006 IberoAmerican Olympiad For University Students, 4

Tags: calculus , integration , algebra , polynomial , function , real analysis

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2019 Jozsef Wildt International Math Competition, W. 62

Tags: integration , inequalities , calculus

Prove that $$\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}$$for all $n\in \mathbb{N}^*$

2007 Croatia Team Selection Test, 1

Tags: calculus , integration , number theory

Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

2010 Today's Calculation Of Integral, 611

Tags: calculus , integration , derivative , function , inequalities , search , logarithm

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

2013 District Olympiad, 1

Tags: limit , integration , calculus , calculus computations

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

2011 Today's Calculation Of Integral, 730

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

Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2021 ISI Entrance Examination, 4

Tags: function , calculus

Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

2022 JHMT HS, 2

Tags: calculus , derivative , function

Suppose that $f$ is a differentiable function such that $f(0) = 20$ and $|f'(x)| \leq 4$ for all real numbers $x$. Let $a$ and $b$ be real numbers such that [i]every[/i] such function $f$ satisfies $a \leq f(22) \leq b$. Find the smallest possible value of $|a| + |b|$.