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

2005 Croatia National Olympiad, 3
Tags: logarithm , inequalities
If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$ \[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]

2007 AMC 12/AHSME, 17
Tags: logarithm
If $ a$ is a nonzero integer and $ b$ is a positive number such that $ ab^{2} \equal{} \log_{10}b,$ what is the median of the set $ \{0,1,a,b,1/b\}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ a \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \frac {1}{b}$

2003 AIME Problems, 4
Tags: trigonometry , logarithm
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 $n$.

1978 IMO Longlists, 28
Tags: function , algebra , logarithm
Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy \[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\] Prove that: $(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$; $(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$). Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.

2022 IMC, 4
Tags: graph coloring , combinatorics , logarithm , set
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.

2005 Today's Calculation Of Integral, 52
Tags: limit , calculus , integration , calculus computations , logarithm
Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]

1997 APMO, 5
Tags: algorithm , rotation , geometry , geometric transformation , combinatorics , logarithm
Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that \[ a_1 + a_2 + \cdots + a_n = nN \] where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.) How should this redistribution be performed so that the total number of objects transferred is minimum?

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

2000 All-Russian Olympiad, 2
Tags: ceiling function , modular arithmetic , number theory , logarithm
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

2006 Romania Team Selection Test, 3
Tags: floor function , function , algebra , logarithm
Let $x_1=1$, $x_2$, $x_3$, $\ldots$ be a sequence of real numbers such that for all $n\geq 1$ we have \[ x_{n+1} = x_n + \frac 1{2x_n} . \] Prove that \[ \lfloor 25 x_{625} \rfloor = 625 . \]

2007 Today's Calculation Of Integral, 205
Tags: calculus , integration , calculus computations , logarithm
Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

2009 Today's Calculation Of Integral, 495
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate the following definite integrals. (1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$ (2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$ (3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$

1980 AMC 12/AHSME, 18
Tags: trigonometry , logarithm
If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals $\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$

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

2007 Moldova National Olympiad, 12.7
Tags: limit , integration , calculus , calculus computations , logarithm
Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

1996 Brazil National Olympiad, 3
Tags: combinatorics , logarithm
Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

1986 India National Olympiad, 2
Tags: algebra , logarithm
Solve \[ \left\{ \begin{array}{l} \log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\ \log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\ \log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\ \end{array} \right.\]

2017 Moscow Mathematical Olympiad, 8
Tags: algebra , logarithm
Are there such $x,y$ that $\lg{(x+y)}=\lg x \lg y$ and $\lg{(x-y)}=\frac{\lg x}{\lg y}$ ?

1976 Miklós Schweitzer, 11
Tags: probability , geometric transformation , reflection , probability and stats , logarithm , geometry
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]

2006 China Second Round Olympiad, 2
Tags: logarithm
Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is ${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\ x>\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $

2011 Kosovo National Mathematical Olympiad, 1
Tags: logarithm
Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers $x^y$, $y^x$ is bigger ?

2013 ISI Entrance Examination, 1
Tags: logarithm
Let $a,b,c$ be real number greater than $1$. Let \[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\] Find the minimum possible value of $S$.

2004 Harvard-MIT Mathematics Tournament, 6
Tags: calculus , logarithm
For $x>0$, let $f(x)=x^x$. Find all values of $x$ for which $f(x)=f'(x)$.

2009 Today's Calculation Of Integral, 444
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

2012 China Team Selection Test, 1
Tags: floor function , function , number theory , logarithm
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]