Olympiad problems

2004 Romania National Olympiad, 3

Tags: logarithms , function , inequalities , calculus , algebra unsolved , algebra

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

2014 NIMO Problems, 6

Tags: logarithms , NIMO , modular arithmetic , number theory , relatively prime , arithmetic sequence

Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]

2002 Putnam, 6

Tags: Putnam , geometry , integration , logarithms , function , college contests , Putnam calculus

Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does \[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} \] converge?

2005 Today's Calculation Of Integral, 1

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

2024 IMC, 2

Tags: logarithms , real analysis , calculus

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

1973 Miklós Schweitzer, 5

Tags: logarithms , limit , integration , real analysis , real analysis unsolved

Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\] [i]P. Medgyessy[/i]

2009 Today's Calculation Of Integral, 441

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2010 Today's Calculation Of Integral, 611

Tags: calculus , integration , derivative , logarithms , function , inequalities , search

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

2013 NIMO Problems, 8

Tags: ceiling function , logarithms , probability , number theory , relatively prime

A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

2002 APMO, 1

Tags: floor function , induction , function , logarithms , inequalities , calculus , derivative

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

2005 Today's Calculation Of Integral, 74

Tags: calculus , integration , logarithms , calculus computations

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

2001 District Olympiad, 4

Tags: limit , integration , logarithms , calculus , real analysis , real analysis unsolved

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

2003 AIME Problems, 4

Tags: trigonometry , logarithms

Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$.

2007 Today's Calculation Of Integral, 255

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

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.

2001 Brazil National Olympiad, 2

Tags: ceiling function , logarithms , number theory , greatest common divisor , relatively prime , number theory proposed

Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers.

2006 Moldova National Olympiad, 10.5

Tags: logarithms , inequalities , calculus , derivative , function , algebra unsolved , algebra

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

2011 Today's Calculation Of Integral, 674

Tags: calculus , integration , logarithms , calculus computations

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]

2011 Today's Calculation Of Integral, 688

Tags: calculus , integration , trigonometry , logarithms , calculus computations

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]

2010 Today's Calculation Of Integral, 641

Tags: calculus , integration , logarithms , calculus computations

Evaluate \[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\] Own

1991 AMC 12/AHSME, 24

Tags: geometry , geometric transformation , rotation , logarithms , AMC

The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$? $ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $

1965 AMC 12/AHSME, 21

Tags: logarithms , limit , function

It is possible to choose $ x > \frac {2}{3}$ in such a way that the value of $ \log_{10}(x^2 \plus{} 3) \minus{} 2 \log_{10}x$ is $ \textbf{(A)}\ \text{negative} \qquad \textbf{(B)}\ \text{zero} \qquad \textbf{(C)}\ \text{one}$ $ \textbf{(D)}\ \text{smaller than any positive number that might be specified}$ $ \textbf{(E)}\ \text{greater than any positive number that might be specified}$

1981 AMC 12/AHSME, 13

Tags: logarithms

Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years, the money will have lost at least $90\%$ of its value. (To the nearest thousandth $\log_{10}3=.477$.) $\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 22$

2002 Moldova National Olympiad, 3

Tags: inequalities , logarithms , function

Let $ a,b> 0$ such that $ a\ne b$. Prove that: $ \sqrt {ab} < \dfrac{a \minus{} b}{\ln a \minus{} \ln b} < \dfrac{a \plus{} b}{2}$

2003 AMC 12-AHSME, 24

Tags: logarithms , trigonometry

If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$? $ \textbf{(A)}\ \minus{}2 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$