Olympiad problems

2001 Romania National Olympiad, 4

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2007 Today's Calculation Of Integral, 222

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

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

2010 Today's Calculation Of Integral, 593

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

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2011 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration

Find all integers $x$ such that $2x^2+x-6$ is a positive integral power of a prime positive integer.

2005 Today's Calculation Of Integral, 48

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

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

2007 Today's Calculation Of Integral, 221

Tags: calculus computations , integration , logarithm , calculus

Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.

2009 Today's Calculation Of Integral, 472

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

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

1969 AMC 12/AHSME, 19

Tags: integration , calculus

The number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$, is: $\textbf{(A) }0\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }12\qquad \textbf{(E) }\text{infinite}$

1991 Arnold's Trivium, 65

Tags: function , integration , geometry , logarithm , domain , 3d geometry , algebra

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

Estonia Open Junior - geometry, 2007.1.4

Tags: ratio , calculus , integration , perimeter , number theory , geometry

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

1998 National Olympiad First Round, 14

Tags: calculus , integration , modular arithmetic

Find the number of distinct integral solutions of $ x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right)$ where $ 0\le x<30$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

2007 Today's Calculation Of Integral, 230

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

Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.

2009 Today's Calculation Of Integral, 424

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

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

2019 Jozsef Wildt International Math Competition, W. 43

Tags: integration , algebra , sequence , limit , polynomial

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

1989 IMO Longlists, 73

Tags: calculus , integration , geometry , trigonometry , symmetry , perimeter , inequalities

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

2005 Today's Calculation Of Integral, 65

Tags: integration , calculus , calculus computations

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

2011 N.N. Mihăileanu Individual, 4

Tags: real analysis , integration , calculus

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.

2024 CMIMC Integration Bee, 13

Tags: integration , calculus

\[\int_0^{2\pi} \frac{1}{3+2 \sqrt{3} \cos x + \cos^2 x}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2011 Today's Calculation Of Integral, 748

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

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

2013 Today's Calculation Of Integral, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2003 District Olympiad, 2

Tags: function , calculus , sequence , integral , ratio , integration

Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as $$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$ [b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $ [b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.

1980 Swedish Mathematical Competition, 4

Tags: integration , inequalities , analysis

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

2010 Today's Calculation Of Integral, 669

Tags: function , integration , calculus , calculus computations

Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$

1998 IberoAmerican Olympiad For University Students, 1

Tags: real analysis , inequalities , calculus , function , integration

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

2007 Today's Calculation Of Integral, 243

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

A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.