Olympiad problems

2018 Romania National Olympiad, 2

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]

2000 Moldova Team Selection Test, 3

Tags: function , polynomial , calculus , probability , trigonometry , integration , algebra

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

2007 Today's Calculation Of Integral, 194

Tags: calculus , integration , calculus computations

Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

2011 Tokio University Entry Examination, 3

Tags: analytic geometry , integration , geometry , limit , perimeter , logarithm , calculus

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2012 Today's Calculation Of Integral, 851

Tags: calculus computations , limit , function , calculus , integration , geometric series , trigonometry

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

1988 IMO Longlists, 28

Tags: modular arithmetic , calculus , polynomial , algebra , integration

Find a necessary and sufficient condition on the natural number $ n$ for the equation \[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0 \] to have a integral root.

2015 AMC 12/AHSME, 23

Tags: calculus , integration , probability , trigonometry , rectangle , geometric probability , geometry

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2010 Today's Calculation Of Integral, 623

Tags: calculus , function , integration , calculus computations

Find the continuous function satisfying the following equation. \[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\] [i]1978 Shibaura Institute of Technology entrance exam[/i]

2014 Taiwan TST Round 1, 1

Tags: integration , function , logarithm

Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.

2007 AIME Problems, 12

Tags: calculus , ratio , logarithm , integration

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

2009 Today's Calculation Of Integral, 454

Tags: trigonometry , calculus , calculus computations , integration

Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

2013 Today's Calculation Of Integral, 891

Tags: rotation , conic , 3d geometry , hyperbola , geometry , calculus , integration

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

1979 Miklós Schweitzer, 8

Tags: integration , function , real analysis

Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent: (i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$. (ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$. [i]V. Totik[/i]

Today's calculation of integrals, 767

Tags: integration , calculus , calculus computations , trigonometry

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2023 CMIMC Integration Bee, 14

Tags: calculus , integration

\[\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2007 Grigore Moisil Intercounty, 1

Tags: calculus , integration , find all functions

Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds. $$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$

2007 Today's Calculation Of Integral, 238

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

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

2007 Harvard-MIT Mathematics Tournament, 33

Tags: calculus , logarithm , integration

Compute \[\int_1^2\dfrac{9x+4}{x^5+3x^2+x}dx.\] (No, your TI-89 doesn’t know how to do this one. Yes, the end is near.)

2005 Today's Calculation Of Integral, 61

Tags: trigonometry , calculus , calculus computations , integration

Evaluate \[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]

2019 Jozsef Wildt International Math Competition, W. 31

Tags: calculus , inequalities , integration

Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

2009 Today's Calculation Of Integral, 517

Tags: geometry , integration , calculus , search , calculus computations

Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.