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

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Found problems: 913

2020 AIME Problems, 3
Tags: logarithm
The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1971 IMO Longlists, 12
Tags: logarithm , algebra
A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that \[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\] Prove that $\prod_{k=1}^n x_k =1.$

Today's calculation of integrals, 897
Tags: geometry , logarithm , rotation , geometric transformation , calculus , integration , calculus computations
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2005 Today's Calculation Of Integral, 15
Tags: calculus , trigonometry , logarithm , integration , calculus computations
Calculate the following indefinite integrals. [1] $\int \frac{(x^2-1)^2}{x^4}dx$ [2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$ [3] $\int \sin 2x\cos 3xdx$ [4] $\int x\ln (x+1)dx$ [5] $\int \frac{x}{(x+3)^2}dx$

1972 AMC 12/AHSME, 8
Tags: logarithm
If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then $\textbf{(A) }x=0\qquad\textbf{(B) }y=1\qquad\textbf{(C) }x=0\text{ and }y=1\qquad$ $\textbf{(D) }x(y-1)=0\qquad \textbf{(E) }\text{None of these}$

2013 Bogdan Stan, 4
Tags: algebra , logarithm , equation
Solve in the real numbers the equation $ 3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1. $ [i]Ion Gușatu[/i]

2003 IMC, 2
Tags: inequalities , trigonometry , logarithm , limit , integration , real analysis , calculus
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

1991 Vietnam Team Selection Test, 3
Tags: matrix , invariant , complex number , logarithm , vector , linear algebra , limit
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

1978 Miklós Schweitzer, 3
Tags: logarithm , number theory , combinatorics
Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

2008 Harvard-MIT Mathematics Tournament, 9
Tags: hmmt , logarithm , calculus computations , integration , real analysis , calculus , limit
([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2010 Today's Calculation Of Integral, 641
Tags: logarithm , integration , calculus computations , calculus
Evaluate \[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\] Own

2007 IberoAmerican Olympiad For University Students, 3
Tags: real analysis , integral inequality , inequalities , logarithm , function , calculus , integration
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

2011 Today's Calculation Of Integral, 763
Tags: logarithm , integration , function , calculus computations , calculus
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$

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 $\log_5 10$? $\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$

1990 AMC 12/AHSME, 23
Tags: logarithm
If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ $ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $

1979 IMO Longlists, 9
Tags: logarithm , inequalities , function
The real numbers $\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n$ are positive. Let us denote by $h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}$ the harmonic mean, $g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}$ the geometric mean, and $a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}$ the arithmetic mean. Prove that $h \leq g \leq a$, and that each of the equalities implies the other one.

2008 AMC 12/AHSME, 16
Tags: arithmetic sequence , logarithm
The numbers $ \log(a^3b^7)$, $ \log(a^5b^{12})$, and $ \log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $ 12^\text{th}$ term of the sequence is $ \log{b^n}$. What is $ n$? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 112 \qquad \textbf{(E)}\ 143$

2009 Indonesia TST, 2
Tags: logarithm , parameterization , inequalities , algebra
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

1976 Miklós Schweitzer, 11
Tags: geometry , geometric transformation , probability and stats , logarithm , reflection , probability
Let $ \xi_1,\xi_2,...$ be independent, identically distributed random variables with distribution \[ P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .\] Write $ S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,$ and \[ T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .\] Prove that $ \liminf_{n \rightarrow \infty} (\log n)T_n=0$ with probability one. [i]P. Revesz[/i]

2004 IMC, 6
Tags: derivative , function , logarithm , integration , algebra , calculus , polynomial
For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

2007 Today's Calculation Of Integral, 216
Tags: calculus computations , logarithm , calculus , integration , limit , function
Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

2009 Princeton University Math Competition, 2
Tags: calculus , ceiling function , logarithm , induction , floor function , integration
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2009 Today's Calculation Of Integral, 399
Tags: logarithm , calculus , integration , calculus computations
Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.

2017 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , logarithm , algebra , inequalities
In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers

1966 AMC 12/AHSME, 16
Tags: logarithm
If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ are real numbers, then $xy$ equals: $\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$