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

2014 USAMO, 6
Tags: logarithm , inequalities
Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]

2000 Croatia National Olympiad, Problem 4
Tags: logarithm , floor function , algebra
If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

2011 Tokyo Instutute Of Technology Entrance Examination, 1
Tags: calculus computations , calculus , limit , logarithm , geometry , trigonometry , integration
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 $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

1988 AIME Problems, 3
Tags: logarithm
Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

2003 China Team Selection Test, 1
Tags: logarithm , function , algebra , induction
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

2010 Today's Calculation Of Integral, 592
Tags: calculus , calculus computations , logarithm , trigonometry , integration
Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

2005 Today's Calculation Of Integral, 17
Tags: logarithm , integration , trigonometry , calculus computations , calculus
Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

2011 AIME Problems, 9
Tags: logarithm , rational root theorem , algebra , trigonometry , polynomial
Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

2010 Korea Junior Math Olympiad, 1
Tags: logarithm , number theory , modular arithmetic
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

1964 AMC 12/AHSME, 1
Tags: logarithm
What is the value of $[\log_{10}(5\log_{10}100)]^2$? ${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $

2019 AMC 12/AHSME, 15
Tags: logarithm
Positive real numbers $a$ and $b$ have the property that \[ \sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100 \] and all four terms on the left are positive integers, where $\text{log}$ denotes the base 10 logarithm. What is $ab$? $\textbf{(A) } 10^{52} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200} $

1958 AMC 12/AHSME, 12
Tags: logarithm
If $ P \equal{} \frac{s}{(1 \plus{} k)^n}$ then $ n$ equals: $ \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad \textbf{(B)}\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad \textbf{(C)}\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \\ \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}$

2009 Today's Calculation Of Integral, 485
Tags: logarithm , function , integration , trigonometry , inequalities , calculus , geometry
In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

2010 AMC 12/AHSME, 24
Tags: logarithm , domain , algebra , floor function , function , trigonometry , polynomial
Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$

1953 AMC 12/AHSME, 22
Tags: logarithm
The logarithm of $ 27\sqrt[4]{9}\sqrt[3]{9}$ to the base $ 3$ is: $ \textbf{(A)}\ 8\frac{1}{2} \qquad\textbf{(B)}\ 4\frac{1}{6} \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these}$

2012 Today's Calculation Of Integral, 853
Tags: calculus computations , logarithm , integration , calculus , trigonometry , limit
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2012 China Northern MO, 6
Tags: logarithm , function , inequalities
Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

2012 Today's Calculation Of Integral, 818
Tags: logarithm , absolute value , function , calculus , integration , limit
For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

2008 Federal Competition For Advanced Students, Part 2, 1
Tags: inequalities , logarithm , function
Prove the inequality \[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3} \] holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.

2005 AIME Problems, 5
Tags: logarithm , floor function
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 $2 \leq b \leq 2005$.

2014 District Olympiad, 3
Tags: logarithm , floor function , algebra
Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set \[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \] has exactly $n+1$ elements.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2
Tags: logarithm , calculus , integration , real analysis , limit
For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2005 AIME Problems, 8
Tags: vieta , algebra , logarithm , number theory , polynomial
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2012 JBMO TST - Turkey, 2
Tags: logarithm , floor function , induction , combinatorics
Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.

2001 District Olympiad, 4
Tags: logarithm , algebra
Solve the equation: \[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\] Note: $\lg x=\log_{10}x$. [i]Daniel Jinga [/i]