Olympiad problems

2023 China Second Round, 2

Tags: logarithm , algebra

if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$

2007 Moldova National Olympiad, 11.2

Tags: limit , logarithm , calculus , calculus computations

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

2010 Today's Calculation Of Integral, 630

Tags: integration , logarithm , calculus , calculus computations

Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$

2004 Romania Team Selection Test, 4

Tags: circumcircle , limit , geometry , logarithm , gauss , polynomial , algebra

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

2012 Today's Calculation Of Integral, 832

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

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

2014 AMC 10, 25

Tags: inequalities , floor function , logarithm , function

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

2005 Today's Calculation Of Integral, 26

Tags: calculus , logarithm , integration , calculus computations

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

1996 Putnam, 2

Tags: geometry , inequalities , induction , rectangle , logarithm , integration

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

2002 AMC 12/AHSME, 21

Tags: logarithm

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$, different from $1$, such that \[2(\log_ac+\log_bc)=9\log_{ab}c.\] Find the largest possible value of $\log_ab$. $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$

2005 AMC 12/AHSME, 17

Tags: number theory , logarithm , greatest common divisor

How many distinct four-tuples $ (a,b,c,d)$ of rational numbers are there with $ a \log_{10} 2 \plus{} b \log_{10} 3 \plus{} c \log_{10} 5 \plus{} d \log_{10} 7 \equal{} 2005$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 2004\qquad \textbf{(E)}\ \text{infinitely many}$

2012 Putnam, 5

Tags: function , logarithm , calculus

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 $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]

1984 All Soviet Union Mathematical Olympiad, 392

Tags: algebra , compare , logarithm

What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).

2014 PUMaC Combinatorics B, 6

Tags: logarithm

Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.

2000 Finnish National High School Mathematics Competition, 3

Tags: logarithm , induction , inequalities , algebra

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

2010 Korea - Final Round, 3

Tags: combinatorics , induction , logarithm , floor function

There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

2006 Iran Team Selection Test, 2

Tags: algebra , logarithm , ceiling function

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

2005 Today's Calculation Of Integral, 21

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

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

2010 Iran Team Selection Test, 1

Tags: number theory , prime number , arithmetic sequence , logarithm , function

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2020 AMC 12/AHSME, 13

Tags: logarithm

Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$ $\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$

1986 AMC 12/AHSME, 25

Tags: floor function , logarithm

If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor = \] $ \textbf{(A)}\ 8192\qquad\textbf{(B)}\ 8204\qquad\textbf{(C)}\ 9218\qquad\textbf{(D)}\ \lfloor \log_{2}(1024!)\rfloor\qquad\textbf{(E)}\ \text{none of these} $

2004 India IMO Training Camp, 1

Tags: function , inequalities , logarithm

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

2007 Junior Tuymaada Olympiad, 5

Tags: logarithm , modular arithmetic , combinatorics , ratio

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

2010 Today's Calculation Of Integral, 540

Tags: calculus , calculus computations , logarithm , integration

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

2013 Korea - Final Round, 3

Tags: inequalities , floor function , logarithm

For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of \[ \sum_{(i,j) \in T} x_i x_j \] in terms of $n$.

2013 ELMO Problems, 2

Tags: function , inequalities , calculus , derivative , 3-variable inequality , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]