Olympiad problems

2010 IMC, 2

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

2007 Today's Calculation Of Integral, 249

Tags: calculus , integration , calculus computations , logarithm

Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2002 District Olympiad, 3

Tags: calculus , integration , real analysis , logarithm

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

2012 Putnam, 5

Tags: function , calculus , logarithm

Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]

2005 Today's Calculation Of Integral, 68

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

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

1974 USAMO, 2

Tags: inequalities , function , logarithm

Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]

2009 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , logarithm

Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2009 Today's Calculation Of Integral, 509

Tags: calculus , integration , trigonometry , algebra , partial fractions , calculus computations , logarithm

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2010 Today's Calculation Of Integral, 666

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

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]

1974 Miklós Schweitzer, 6

Tags: induction , function , integration , limit , logarithm , real analysis

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

2011 Today's Calculation Of Integral, 724

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

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2005 France Team Selection Test, 4

Tags: floor function , ceiling function , induction , inequalities , number theory , logarithm

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2014 Junior Balkan MO, 4

Tags: floor function , modular arithmetic , limit , combinatorics , logarithm

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2009 Today's Calculation Of Integral, 439

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

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

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 $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

2000 Baltic Way, 20

Tags: integration , trigonometry , algebra , logarithm

For every positive integer $n$, let \[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\] Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.

2011 Today's Calculation Of Integral, 748

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

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , logarithm

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

2003 CentroAmerican, 3

Tags: calculus , induction , logarithm , inequalities

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

2007 Today's Calculation Of Integral, 255

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

Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.

1955 AMC 12/AHSME, 17

Tags: logarithm

If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals: $ \textbf{(A)}\ 1.25 \qquad \textbf{(B)}\ 0.81 \qquad \textbf{(C)}\ 2.43 \qquad \textbf{(D)}\ 0.8 \qquad \textbf{(E)}\ \text{either 0.8 or 1.25}$

2009 Today's Calculation Of Integral, 434

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

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

1978 IMO Longlists, 37

Tags: algebra , logarithm

Simplify \[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\] where $a, b, c$ are positive real numbers.

1950 AMC 12/AHSME, 44

Tags: logarithm

The graph of $ y\equal{}\log x$ $\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ \textbf{(D)}\ \text{Cuts neither axis} \qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$

2003 Pan African, 1

Tags: function , induction , logarithm

Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that: (1) $f(n) < f(n+1)$, all $n \in N_0$; (2) $f(2)=2$; (3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.