Olympiad problems

2010 Today's Calculation Of Integral, 665

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

2006 International Zhautykov Olympiad, 2

Tags: calculus , inequalities

Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality: \[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd). \]

2002 VJIMC, Problem 4

Tags: integration , calculus

Prove that $$\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.$$

1999 Canada National Olympiad, 5

Tags: three variable inequality , calculus , algebra , inequalities

Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x \plus{} y \plus{} z \equal{} 1$. Show that \[ x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27} \] and find when equality occurs.

2010 Tuymaada Olympiad, 2

Tags: circumcircle , parallelogram , trigonometry , reflection , calculus , geometric transformation , geometry

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

2010 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , geometry

Let $f(x)=x^3-x^2$. For a given value of $x$, the graph of $f(x)$, together with the graph of the line $c+x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.

2012 Today's Calculation Of Integral, 812

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

Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$

2012 Today's Calculation Of Integral, 835

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

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

2005 Today's Calculation Of Integral, 10

Tags: calculus computations , calculus , trigonometry , integration

Calculate the following indefinite integrals. [1] $\int (2x+1)\sqrt{x+2}\ dx$ [2] $\int \frac{1+\cos x}{x+\sin x}\ dx$ [3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$ [4] $\int \frac{(x-3)^2}{x^4}\ dx$ [5] $\int \frac{dx}{\tan x}\ dx$

2013 Online Math Open Problems, 25

Tags: geometric transformation , perimeter , calculus , geometry , function , inequalities

Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. [i]Proposed by Evan Chen[/i]

2011 Today's Calculation Of Integral, 690

Tags: calculus computations , calculus , integration , vector , trigonometry

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

2004 Harvard-MIT Mathematics Tournament, 9

Tags: algebra , absolute value , logarithm , polynomial , calculus , limit , ratio

Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.

2002 VJIMC, Problem 3

Tags: calculus , integration , inequalities

Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying $$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by $$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.

2009 Today's Calculation Of Integral, 469

Tags: calculus , calculus computations , trigonometry , integration

Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.

2014 VJIMC, Problem 4

Tags: inequalities , integration , calculus

Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that $$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$

1985 Traian Lălescu, 2.2

Tags: inequalities , calculus , young s inequality , integral inequality , bijective , integral , real analysis

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

2009 AIME Problems, 11

Tags: calculus , absolute value , integration , geometry , vector , analytic geometry

Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.

2005 Today's Calculation Of Integral, 5

Tags: integration , trigonometry , calculus , calculus computations

Calculate the following indefinite integrals. [1] $\int (4-5\tan x)\cos x dx$ [2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$ [3] $\int x^3\sqrt{4-x^2}dx$ [4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$ [5] $\int (3x-4)^2 dx$

2024 CMIMC Integration Bee, 9

Tags: integration , calculus

\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2012 Today's Calculation Of Integral, 831

Tags: calculus computations , limit , integration , calculus , induction

Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

2010 Today's Calculation Of Integral, 585

Tags: calculus computations , logarithm , integration , calculus

Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

2004 USAMO, 3

Tags: quadratic , calculus , function , ratio , geometry

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

2008 Harvard-MIT Mathematics Tournament, 3

Tags: logarithm , calculus , integration , calculus computations

([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.

2011 India National Olympiad, 3

Tags: polynomial , integration , rational root theorem , calculus , algebra , number theory

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$

2005 Today's Calculation Of Integral, 88

Tags: calculus computations , calculus , function , integration

A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$ Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$