Olympiad problems

1997 IMC, 1

Tags: limit , calculus , integration , logarithm , real analysis

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

2005 Today's Calculation Of Integral, 59

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

1969 AMC 12/AHSME, 27

Tags: calculus , integration

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: $\textbf{(A) }\dfrac2{n-1}\qquad \textbf{(B) }\dfrac{n-1}2\qquad \textbf{(C) }\dfrac2n\qquad \textbf{(D) }2n\qquad \textbf{(E) }2(n-1)$

2004 Romania National Olympiad, 3

Tags: function , integration , calculus , inequalities , polynomial , linear algebra , algebra

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

2011 All-Russian Olympiad, 3

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

Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

2014 Contests, 2

Tags: inequalities , function , integration , combinatorics

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

2015 Romania National Olympiad, 4

Tags: algebra , polynomial , function , calculus , integration

Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $

1989 IMO Longlists, 76

Tags: calculus , integration , combinatorics

Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth $ n$ bourbakis and each silver coin is worth $ m$ bourbakis ($ n$ and $ m$ are positive integers). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. But Poldavia’s monetary system is not as strange as it seems: [b](a)[/b] Prove that it is possible to buy anything that costs an integral number of bourbakis, as long as one can receive change. [b](b)[/b] Prove that any payment above $ mn\minus{}2$ bourbakis can be made without the need to receive change.

2019 Ramnicean Hope, 2

Tags: calculus , integration , definite integral , quadratic

Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $ [i]Ovidiu Țâțan[/i]

2008 Grigore Moisil Intercounty, 3

Tags: function , inequalities , calculus , integration

Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $ [b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $ [b]b)[/b] Determine $ f $ if the above inequality is actually an equality. [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2007 District Olympiad, 2

Tags: calculus , integration , inequalities , function , derivative , probability , expected value

Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$. Prove that if $f$ is increasing, then \[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]

2010 Today's Calculation Of Integral, 550

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

2011 Today's Calculation Of Integral, 710

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

Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2013 Today's Calculation Of Integral, 871

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

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 $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2013 Hitotsubashi University Entrance Examination, 3

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

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

1999 Finnish National High School Mathematics Competition, 1

Tags: calculus , integration , modular arithmetic , number theory

Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$

2013 Today's Calculation Of Integral, 895

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

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.