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: 894

1953 AMC 12/AHSME, 5
Tags: logarithms
If $ \log_6 x\equal{}2.5$, the value of $ x$ is: $ \textbf{(A)}\ 90 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 36\sqrt{6} \qquad\textbf{(D)}\ 0.5 \qquad\textbf{(E)}\ \text{none of these}$

1984 IMO Shortlist, 20
Tags: inequalities , logarithms , algebra , IMO Shortlist
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2011 Pre-Preparation Course Examination, 3
Tags: logarithms , advanced fields , advanced fields unsolved
prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$

2013 NIMO Problems, 7
Tags: logarithms
For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer. [i]Proposed by Evan Chen[/i]

2018 District Olympiad, 3
Tags: logarithms , inequalities
Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and \[\log_ab + 2\log_bc + 5\log_ca = 12.\] Prove that \[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]

1978 IMO Longlists, 37
Tags: logarithms , algebra proposed , algebra
Simplify \[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\] where $a, b, c$ are positive real numbers.

2013 ISI Entrance Examination, 1
Tags: logarithms
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 $S$.

2008 Putnam, A4
Tags: Putnam , logarithms , integration , floor function , Putnam calculus
Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?

Today's calculation of integrals, 860
Tags: calculus , integration , function , logarithms , analytic geometry , geometry , derivative
For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

2006 Iran Team Selection Test, 2
Tags: ceiling function , logarithms , algebra proposed , algebra
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) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

1961 AMC 12/AHSME, 6
Tags: logarithms , AMC
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: ${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $

2012 Iran MO (3rd Round), 3
Tags: function , ceiling function , logarithms , induction , number theory proposed , number theory
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. [i]Proposed by Amirhossein Gorzi[/i]

1984 AMC 12/AHSME, 9
Tags: logarithms , floor function , number theory , prime factorization
The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27

2006 AMC 12/AHSME, 21
Tags: geometry , logarithms , ratio , inequalities , function , AMC
Let \[ S_1 \equal{} \{ (x,y)\ | \ \log_{10} (1 \plus{} x^2 \plus{} y^2)\le 1 \plus{} \log_{10}(x \plus{} y)\} \]and \[ S_2 \equal{} \{ (x,y)\ | \ \log_{10} (2 \plus{} x^2 \plus{} y^2)\le 2 \plus{} \log_{10}(x \plus{} y)\}. \]What is the ratio of the area of $ S_2$ to the area of $ S_1$? $ \textbf{(A) } 98\qquad \textbf{(B) } 99\qquad \textbf{(C) } 100\qquad \textbf{(D) } 101\qquad \textbf{(E) } 102$

2004 IMC, 6
Tags: function , logarithms , algebra , polynomial , calculus , derivative , integration
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 \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

1966 AMC 12/AHSME, 24
Tags: logarithms
If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals: $\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$

2004 China Western Mathematical Olympiad, 4
Tags: function , Euler , search , logarithms , number theory , totient function , prime numbers
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2006 Hanoi Open Mathematics Competitions, 3
Tags: algebra , logarithms
Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$. Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .

2020 IMC, 8
Tags: IMC , real analysis , binomial coefficients , logarithms , IMC 2020
Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

1966 AMC 12/AHSME, 16
Tags: logarithms
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: $\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$

2006 AMC 12/AHSME, 20
Tags: probability , floor function , logarithms , function , calculus , integration , inequalities
Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

2005 Today's Calculation Of Integral, 8
Tags: calculus , integration , logarithms , trigonometry , function , calculus computations
Calculate the following indefinite integrals. [1] $\int x(x^2+3)^2 dx$ [2] $\int \ln (x+2) dx$ [3] $\int x\cos x dx$ [4] $\int \frac{dx}{(x+2)^2}dx$ [5] $\int \frac{x-1}{x^2-2x+3}dx$

2021 AMC 12/AHSME Fall, 9
Tags: logarithms
A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_2 x$, $\log_3 x$, and $\log_4 x$. What is $x$? $\textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }6\sqrt{6}\qquad\textbf{(C) }24\qquad\textbf{(D) }48\qquad\textbf{(E) }576$

2012 Online Math Open Problems, 28
Tags: Online Math Open , modular arithmetic , real analysis , function , logarithms , limit , number theory
Find the remainder when \[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.) [i]Victor Wang.[/i]

2007 Today's Calculation Of Integral, 230
Tags: calculus , integration , logarithms , function , calculus computations
Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.