Olympiad problems

2009 Today's Calculation Of Integral, 454

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.

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 $\int_0^1 f(t)dt.$

2005 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function

Let $ f : \mathbf{R} \to \mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.

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.)

1999 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , geometry , rectangle , trigonometry

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

1993 Vietnam National Olympiad, 1

Tags: calculus , real analysis , algebra , trigonometry

$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.

2005 Today's Calculation Of Integral, 61

Tags: trigonometry , calculus , calculus computations , integration

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

2014 AMC 12/AHSME, 25

Tags: vector , derivative , parabola , parameterization , analytic geometry , calculus , conic

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

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 $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

ICMC 5, 2

Tags: calculus , analysis , algebra

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

1992 IMO Longlists, 74

Tags: logarithm , calculus , algebra , limit , pigeonhole principle

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

2008 Czech and Slovak Olympiad III A, 1

Tags: number theory , modular arithmetic , calculus

In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

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$.

1992 AIME Problems, 13

Tags: inequalities , function , calculus , quadratic , analytic geometry , geometry , ratio

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

2008 Estonia Team Selection Test, 3

Tags: inequalities , function , calculus , algebra

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2007 Today's Calculation Of Integral, 255

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

Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.

2011 Federal Competition For Advanced Students, Part 1, 1

Tags: calculus , modular arithmetic , integration , quadratic , prime factorization , number theory

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

2005 Gheorghe Vranceanu, 4

Tags: calculus , cesaro-stolz

$ \lim_{n\to\infty } \left( (1+1/n)^{-n}\sum_{i=0}^n\frac{1}{i!} \right)^{2n} $

2005 ISI B.Math Entrance Exam, 1

Tags: integration , calculus

For any $k\in\mathbb{Z}^+$ , prove that:- $2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1})$ Also compute integral part of $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}$.

2007 Princeton University Math Competition, 4

Tags: integration , calculus

Find the sum of the reciprocals of the positive integral factors of $84$.

PEN P Problems, 12

Tags: calculus , integration , inequalities , floor function , function , logarithm

The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]