Olympiad problems

2005 Today's Calculation Of Integral, 58

Let $f(x)=\frac{e^x}{e^x+1}$ Prove the following equation. \[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]

2013 ELMO Shortlist, 5

Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

Today's calculation of integrals, 849

Tags: calculus , integration , function , calculus computations

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2018 PUMaC Live Round, Calculus 2

Tags: calculus , probability

Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between $1\!:\!00$ pm and $2\!:\!00$ pm. Each friend will only wait for $5$ minutes by themselves before leaving. However, if another friend arrives within those $5$ minutes, the pair will wait $15$ minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as $\tfrac{m}{n}$, what is $m+n$?

2007 Estonia Math Open Junior Contests, 4

Tags: calculus , integration , ratio , 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.

2010 Today's Calculation Of Integral, 649

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

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

2006 ISI B.Math Entrance Exam, 4

Tags: calculus , derivative , function , induction , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2005 Today's Calculation Of Integral, 77

Tags: calculus , integration , geometry , calculus computations

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

2007 Today's Calculation Of Integral, 171

Tags: calculus , integration , calculus computations

Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$

1989 Putnam, B3

Tags: calculus , integration , de , differential equation

Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy $$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define $$\mu_n=\int^\infty_0x^nf(x)dx.$$ $a.$ Express $\mu_n$ in terms of $\mu_0$. $b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.

2009 Today's Calculation Of Integral, 467

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

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

2007 Today's Calculation Of Integral, 197

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

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

2022 CMIMC Integration Bee, 1

Tags: calculus , integration

\[\int_0^{\pi/1011}\sin^2(2022x)+\cos^2(2022x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

1967 Putnam, B6

Tags: function , calculus , derivative , partial derivatives

Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that $$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$

2005 Today's Calculation Of Integral, 13

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

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

1984 Vietnam National Olympiad, 1

Tags: polynomial , calculus , integration , algebra

$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root. $(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.

2005 Today's Calculation Of Integral, 60

Tags: calculus , integration , trigonometry , calculus computations

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2009 Moldova Team Selection Test, 2

Tags: function , calculus , derivative , rearrangement inequality , logarithm , inequalities

[color=darkred]Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i \equal{} \overline{1,n}$, such that $ \sum a_i \equal{} 1$. Prove that $ \small{\dfrac{a_1^{2 \minus{} m} \plus{} a_2 \plus{} ... \plus{} a_{n \minus{} 1}}{1 \minus{} a_1} \plus{} \dfrac{a_2^{2 \minus{} m} \plus{} a_3 \plus{} ... \plus{} a_n}{1 \minus{} a_1} \plus{} ... \plus{} \dfrac{a_n^{2 \minus{} m} \plus{} a_1 \plus{} ... \plus{} a_{n \minus{} 2}}{1 \minus{} a_1}\ge n \plus{} \dfrac{n^m \minus{} n}{n \minus{} 1}}$[/color]

2017 Azerbaijan EGMO TST, 4

Tags: calculus , integration , induction , modular arithmetic , number theory

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2015 Brazil Team Selection Test, 3

Tags: algebra , function , calculus

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

2005 Today's Calculation Of Integral, 81

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

Prove the following inequality. \[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]

2020 Jozsef Wildt International Math Competition, W23

Tags: integration , calculus

Prove that $$\int^{\frac\pi3}_{\frac\pi6}\frac u{\sin u}du=\frac83\sum_{k=0}^\infty\frac{(-1)^k}{3^k(2k+1)^2}+\frac{\pi\ln3}{3\sqrt3}-\frac{4C}3+\frac\pi6\ln\left(2+\sqrt3\right)-\operatorname{Im}\left(\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)-\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ and $C=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}$ is the Catalan constant. [i]Proposed by Paolo Perfetti[/i]

1976 Miklós Schweitzer, 4

Tags: calculus , integration , algebra , function , domain , polynomial , ring theory

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

2015 Vietnam National Olympiad, 1

Tags: limit , calculus , algebra

Given a non negative real $a$ and a sequence $(u_n)$ defined by \[ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} \] a) Prove that for $a=0$, the sequence is convergent and find its limit. b) For $a\in [0,1]$, prove that the sequence if convergent.

2009 Today's Calculation Of Integral, 519

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.