Olympiad problems

2014 Online Math Open Problems, 29

Tags: 3d geometry , tetrahedron , geometry , calculus , circumcircle , integration

Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$. [i]Proposed by Sammy Luo and Evan Chen[/i]

2007 Gheorghe Vranceanu, 4

Tags: real analysis , cesaro-stolz , sequence , integral limit , function , integration , calculus

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$

2007 Harvard-MIT Mathematics Tournament, 9

Tags: limit , calculus , function

$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align} Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.

2011 Today's Calculation Of Integral, 727

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

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

2011 Today's Calculation Of Integral, 702

Tags: calculus computations , calculus , integration , function

$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$

1989 Putnam, B6

Tags: expected value , probability , integration , calculus

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 $0\le t\le1$.

2005 Today's Calculation Of Integral, 68

Tags: calculus , logarithm , calculus computations , derivative , integration

Find the minimum value of $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

2005 Harvard-MIT Mathematics Tournament, 7

Tags: calculus

Let $x$ be a positive real number. Find the maximum possible value of \[\frac{x^2+2-\sqrt{x^4+4}}{x}.\]

1989 IMO Shortlist, 8

Tags: calculus , rectangle , integration , geometry , number theory

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: derivative , geometry , trigonometry , inequalities , calculus

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

2014 NIMO Problems, 3

Tags: calculus , integration

In land of Nyemo, the unit of currency is called a [i]quack[/i]. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay oﬀ $125$ quacks using these coins? [i]Proposed by Aaron Lin[/i]

1972 Canada National Olympiad, 10

Tags: calculus , geometric sequence , integration , ratio , algebra

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2010 Today's Calculation Of Integral, 553

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

Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.

2021 JHMT HS, 2

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

Compute the smallest positive integer $n$ such that $\int_{0}^{n} \lfloor x\rfloor\,dx$ is at least $2021.$

2009 Today's Calculation Of Integral, 505

Tags: calculus computations , trigonometry , integration , calculus

In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.

2006 Harvard-MIT Mathematics Tournament, 10

Tags: logarithm , function , integration , calculus

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

2011 Today's Calculation Of Integral, 735

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

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2009 Today's Calculation Of Integral, 498

Tags: calculus computations , integration , function , calculus

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

Today's calculation of integrals, 859

Tags: geometry , calculus , derivative , integration , calculus computations

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2022 VJIMC, 3

Tags: calculus , function , integration

Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit $$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$

2010 Morocco TST, 2

Tags: algebra , series summation , approximation , floor function , integration , calculus

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

1996 Vietnam National Olympiad, 3

Tags: derivative , calculus , polynomial , algebra , inequalities

Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

2007 Harvard-MIT Mathematics Tournament, 5

Tags: function , calculus

The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.

2007 Today's Calculation Of Integral, 180

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

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2009 Today's Calculation Of Integral, 480

Tags: integration , inequalities , calculus , calculus computations , geometry

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]