This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 913

1979 AMC 12/AHSME, 18
Tags: logarithm
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$? $\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$

2009 China Second Round Olympiad, 2
Tags: inequalities , logarithm
Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

2013 Korea - Final Round, 3
Tags: floor function , inequalities , 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 $n$.

2016 Nigerian Senior MO Round 2, Problem 10
Tags: algebra , logarithm
Positive numbers $x$ and $y$ satisfy $xy=2^{15}$ and $\log_2{x} \cdot \log_2{y} = 60$. Find $\sqrt[3]{(\log_2{x})^3+(\log_2{y})^3}$

2013 AMC 12/AHSME, 21
Tags: calculus , trigonometry , logarithm
Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $? $ \textbf{(A)} \ (\log 2016, \log 2017) $ $ \textbf{(B)} \ (\log 2017, \log 2018) $ $ \textbf{(C)} \ (\log 2018, \log 2019) $ $ \textbf{(D)} \ (\log 2019, \log 2020) $ $ \textbf{(E)} \ (\log 2020, \log 2021) $

2000 Finnish National High School Mathematics Competition, 3
Tags: inequalities , induction , algebra , logarithm
Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

2011 Today's Calculation Of Integral, 694
Tags: calculus , integration , inequalities , quadratic , calculus computations , logarithm
Prove the following inequality: \[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\] created by kunny

1966 IMO Shortlist, 30
Tags: inequalities , calculus , logarithm
Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2010 AMC 12/AHSME, 12
Tags: logarithm
For what value of $ x$ does \[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\] $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$

2014 NIMO Problems, 6
Tags: ceiling function , floor function , probability , expected value , number theory , relatively prime , logarithm
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

2012 Today's Calculation Of Integral, 824
Tags: calculus , integration , geometry , limit , calculus computations , logarithm
In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

2018 District Olympiad, 1
Tags: logarithm
Find $x\in\mathbb{R}$ for which \[\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.\]

2024 AIME, 2
Tags: logarithm
Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?

2011 Today's Calculation Of Integral, 743
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

2005 Today's Calculation Of Integral, 21
Tags: calculus , integration , trigonometry , calculus computations , logarithm
[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.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

1995 Turkey MO (2nd round), 3
Tags: function , algebra , logarithm
Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\] $(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$. $(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.

2011 Today's Calculation Of Integral, 737
Tags: calculus , integration , geometry , inequalities , calculus computations , logarithm
Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

2002 District Olympiad, 3
Tags: number theory , inequalities , logarithm
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $ Show that $ a<b. $

2014 ELMO Shortlist, 9
Tags: number theory , logarithm
Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

2009 Putnam, A2
Tags: function , calculus , derivative , trigonometry , integration , logarithm
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

2011 Kazakhstan National Olympiad, 1
Tags: logarithm , algebra
Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $

2011 Today's Calculation Of Integral, 736
Tags: calculus , integration , derivative , calculus computations , logarithm
Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

2009 Harvard-MIT Mathematics Tournament, 2
Tags: hmmt , complex number , logarithm
Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

2005 AIME Problems, 5
Tags: floor function , logarithm
Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.

1981 AMC 12/AHSME, 15
Tags: logarithm
If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$