Olympiad problems

2013 Online Math Open Problems, 25

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

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]

2006 Petru Moroșan-Trident, 2

Tags: indefinite integral , calculus

Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $ [i]Dan Negulescu[/i]

2011 Tokyo Instutute Of Technology Entrance Examination, 1

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

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

1989 Putnam, B6

Tags: probability , expected value , calculus , integration

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

1993 Vietnam National Olympiad, 1

Tags: calculus , trigonometry , real analysis , algebra

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

2009 Today's Calculation Of Integral, 468

Tags: calculus , integration , function , calculus computations

Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

PEN I Problems, 8

Tags: floor function , calculus , derivative , inequalities

Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.

1995 Putnam, 2

Tags: calculus , integration

For what pairs of positive real numbers $(a,b)$ does the improper integral $(1)$ converge? \begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}

2009 Today's Calculation Of Integral, 454

Tags: calculus , integration , trigonometry , calculus computations

Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

Today's calculation of integrals, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2019 ISI Entrance Examination, 4

Tags: calculus

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0$$ Show that there exist $a,b\in\mathbb{R}$ such that $f(x)=ax+b$ for all $x\in\mathbb{R}$.

2012 Today's Calculation Of Integral, 851

Tags: calculus , integration , function , trigonometry , limit , geometric series , calculus computations

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2014 India Regional Mathematical Olympiad, 2

Tags: calculus , integration , arithmetic sequence , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2005 Postal Coaching, 24

Tags: calculus , integration , number theory

Find all nonnegative integers $x,y$ such that \[ 2 \cdot 3^{x} +1 = 7 \cdot 5^{y}. \]

1950 Miklós Schweitzer, 7

Tags: integration , calculus , euler , function , logarithm , real analysis

Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

2011 Today's Calculation Of Integral, 759

Tags: calculus , integration , 3d geometry , tetrahedron , geometric transformation , rotation , geometry

Given a regular tetrahedron $PQRS$ with side length $d$. Find the volume of the solid generated by a rotation around the line passing through $P$ and the midpoint $M$ of $QR$.

2010 Today's Calculation Of Integral, 606

Tags: calculus , integration , geometry , calculus computations

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

2009 Serbia Team Selection Test, 1

Tags: inequalities , trigonometry , calculus , derivative , angle bisector , geometry

Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.

2010 Contests, 1

Tags: number theory , greatest common divisor , polynomial , modular arithmetic , induction , calculus , algebra

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

1998 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , search , trigonometry , function

Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$. Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$. What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?

1989 APMO, 5

Tags: function , induction , calculus , derivative , limit , algebra , recursion

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)

2005 MOP Homework, 5

Tags: function , integration , calculus , binomial theorem , algebra

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

1990 Flanders Math Olympiad, 4

Tags: function , limit , calculus , calculus computations

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function. (a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$ (b) Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.

1952 AMC 12/AHSME, 1

Tags: geometry , calculus , integration

If the radius of a circle is a rational number, its area is given by a number which is: $ \textbf{(A)}\ \text{rational} \qquad\textbf{(B)}\ \text{irrational} \qquad\textbf{(C)}\ \text{integral} \qquad\textbf{(D)}\ \text{a perfect square}$ $ \textbf{(E)}\ \text{none of these}$