Olympiad problems

Today's calculation of integrals, 764

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

Find $\lim_{n\to\infty} \int_0^{\pi} e^{x}|\sin nx|dx.$

2012 Today's Calculation Of Integral, 790

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

Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$. (1) Find the equations of the tangents $l_1,\ l_2$. (2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.

2024 VJIMC, 1

Tags: inequalities , function , calculus , integration , derivative

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that \[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]

2012 Today's Calculation Of Integral, 808

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

For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\lim_{n\to\infty} a_n$.

1984 Putnam, B4

Tags: function , integration , calculus

Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$, positive on $(0,\infty)$, such that for all $x>0$ the $y$-coordinate of the centroid of the region $$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$is the same as the average value of $g$ on $[0,x]$.

1971 IMO Longlists, 47

Tags: calculus , integration , real analysis , algebra

A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?

2007 Today's Calculation Of Integral, 237

Tags: calculus , integration , absolute value , calculus computations , logarithm

Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

2012 Today's Calculation Of Integral, 773

Tags: calculus , integration , calculus computations , logarithm

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

2011 Today's Calculation Of Integral, 680

Tags: calculus , integration , calculus computations

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2011 Mediterranean Mathematics Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , sphere , trigonometry , integration

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

2007 Moldova National Olympiad, 12.7

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

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

1991 Arnold's Trivium, 1

Tags: calculus , derivative , integration , function , analytic geometry , graphing lines , slope

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

2008 Iran MO (3rd Round), 1

Tags: algebra , polynomial , trigonometry , function , calculus , integration , ratio

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle

Today's calculation of integrals, 850

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]

2002 AMC 12/AHSME, 24

Tags: trigonometry , geometry , calculus , integration , modular arithmetic

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

1995 Brazil National Olympiad, 5

Tags: algebra , polynomial , calculus , integration , group theory , abstract algebra , galois theory

Show that no one $n$-th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.

2011 Today's Calculation Of Integral, 688

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

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

1995 IberoAmerican, 3

Tags: function , floor function , induction , calculus , integration , algebra

A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function. [b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.

2011 Putnam, B5

Tags: integration , function , inequalities , calculus , vector

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

2010 Today's Calculation Of Integral, 668

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

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin　University entrance exam/Medicine [/i]

1991 Arnold's Trivium, 52

Tags: calculus , integration , function

Calculate the first term of the asymptotic expression as $k\to\infty$ of the integral \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx\]

2010 Today's Calculation Of Integral, 524

Tags: calculus , integration , function , calculus computations

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: calculus , integration , limit , logarithm , real analysis

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2003 IMC, 2

Tags: calculus , integration , limit , trigonometry , inequalities , real analysis , logarithm

Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

2012 Today's Calculation Of Integral, 833

Tags: calculus , integration , trigonometry , derivative , complex number , calculus computations

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$