Olympiad problems

2008 Moldova Team Selection Test, 4

Tags: calculus , polynomial , factorial , algebra , integration

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

2012 Today's Calculation Of Integral, 857

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

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2009 Today's Calculation Of Integral, 432

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

Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.

1963 Miklós Schweitzer, 10

Tags: geometric transformation , combinatorial geometry , analytic geometry , rotation , probability , integration , geometry

Select $ n$ points on a circle independently with uniform distribution. Let $ P_n$ be the probability that the center of the circle is in the interior of the convex hull of these $ n$ points. Calculate the probabilities $ P_3$ and $ P_4$. [A. Renyi]

2010 Abels Math Contest (Norwegian MO) Final, 4b

Tags: calculus , number theory , integration

Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).

1966 Miklós Schweitzer, 10

Tags: integration , probability and stats

For a real number $ x$ in the interval $ (0,1)$ with decimal representation \[ 0.a_1(x)a_2(x)...a_n(x)...,\] denote by $ n(x)$ the smallest nonnegative integer such that \[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\] Determine $ \int_0^1n(x)dx$. ($ \overline{abcd}$ denotes the decimal number with digits $ a,b,c,d .$) [i]A. Renyi[/i]

2011 Romania National Olympiad, 4

Tags: function , integration , real analysis

Let $ f,F:\mathbb{R}\longrightarrow\mathbb{R} $ be two functions such that $ f $ is nondecreasing, $ F $ admits finite lateral derivates in every point of its domain, $$ \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , $$ for all real numbers $ y, $ and $ F(0)=0. $ Prove that $ F(x)=\int_0^x f(t)dt, $ for all real numbers $ x. $

2013 Today's Calculation Of Integral, 865

Tags: rectangle , calculus , geometric transformation , integration , analytic geometry , rotation , geometry

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

2005 Today's Calculation Of Integral, 43

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

Evaluate \[\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx\]

2013 Today's Calculation Of Integral, 880

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

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2009 Putnam, A4

Tags: modular arithmetic , calculus , induction , greatest common divisor , quadratic , integration

Let $ S$ be a set of rational numbers such that (a) $ 0\in S;$ (b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and (c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$ Must $ S$ contain all rational numbers?

2010 Romania National Olympiad, 4

Tags: function , limit , derivative , real analysis , integration , calculus

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

2012 Today's Calculation Of Integral, 803

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

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2007 Today's Calculation Of Integral, 204

Tags: trigonometry , integration , special factorizations , calculus , calculus computations

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

2007 Today's Calculation Of Integral, 210

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

Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.

2014 Indonesia MO Shortlist, A6

Tags: polynomial , calculus , algebra , integration

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)

2009 Today's Calculation Of Integral, 491

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

Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ (1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$. (2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$

2005 Today's Calculation Of Integral, 12

Tags: calculus , integration , trigonometry , calculus computations

Calculate the following indefinite integrals. [1] $\int \frac{dx}{1+\cos x}$ [2] $\int x\sqrt{x^2-1}dx$ [3] $\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)$ [4] $\int \frac{\sin ^ 3 x}{1+\cos x}dx$ [5] $\int e^{4x}\sin 2x dx$

2010 Contests, 523

Tags: logarithm , integration , calculus computations , calculus

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2010 Today's Calculation Of Integral, 625

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

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2010 Today's Calculation Of Integral, 599

Tags: trigonometry , integration , calculus , calculus computations

Evaluate $\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx$. created by kunny

2007 Today's Calculation Of Integral, 225

Tags: calculus , logarithm , geometry , function , integration , ratio , inequalities

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

2010 Today's Calculation Of Integral, 570

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

Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$. (1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$. (2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$. (3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.

2010 Today's Calculation Of Integral, 666

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

Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying: (i) $f\left(\frac{\pi}{6}\right)=0$ (ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$. Find $f(x)$. [i]1987 Sapporo Medical University entrance exam[/i]

2006 Hong Kong TST., 1

Tags: integration , calculus

Find the integral solutions of the equation $7(x+y)=3(x^2-xy+y^2)$