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: 894

2012 IberoAmerican, 3
Tags: logarithms , floor function , number theory proposed , number theory
Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits. (Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)

1973 AMC 12/AHSME, 28
Tags: logarithms , ratio , geometric sequence , arithmetic sequence , arithmetic series
If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence $ \textbf{(A)}\ \text{which is a G.P} \qquad$ $ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$ $ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$ $ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$ $ \textbf{(E)}\ \text{none of these}$

1994 China Team Selection Test, 1
Tags: inequalities , logarithms , function , algebra , domain , linear algebra , matrix
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

1950 AMC 12/AHSME, 44
Tags: logarithms
The graph of $ y\equal{}\log x$ $\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ \textbf{(D)}\ \text{Cuts neither axis} \qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$

1975 AMC 12/AHSME, 19
Tags: logarithms
Which positive numbers $ x$ satisfy the equation $ (\log_3x)(\log_x5)\equal{}\log_35$? $ \textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad$ $ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are }$ $ \text{positive integers} \qquad$ $ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad \textbf{(E)}\ \text{none of these}$

1995 Turkey MO (2nd round), 3
Tags: function , logarithms , algebra unsolved , algebra
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 $n\in \mathbb{N}$.

1983 AMC 12/AHSME, 25
Tags: logarithms , AoPS Books
If $60^a = 3$ and $60^b = 5$, then $12^{[(1-a-b)/2(1-b)]}$ is $\text{(A)} \ \sqrt{3} \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt{5} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \sqrt{12}$

2005 Today's Calculation Of Integral, 38
Tags: calculus , integration , logarithms , calculus computations
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

1998 National Olympiad First Round, 2
Tags: floor function , logarithms
Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$ $\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$

2011 Today's Calculation Of Integral, 710
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2014 PUMaC Combinatorics B, 6
Tags: logarithms
Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.

2014 Albania Round 2, 4
Tags: number theory , algebra , trigonometry , logarithms
Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

2009 Today's Calculation Of Integral, 452
Tags: calculus , integration , logarithms , calculus computations
Let $ a,\ b$ are postive constant numbers. (1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$ (2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$

2011 District Olympiad, 4
Tags: logarithms , algebra , a implies b
[b]a)[/b] Show that , if $ a,b>1 $ are two distinct real numbers, then $ \log_a\log_a b >\log_b\log_a b. $ [b]b)[/b] Show that if $ a_1>a_2>\cdots >a_n>1 $ are $ n\ge 2 $ real numbers, then $$ \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0. $$

2010 Today's Calculation Of Integral, 627
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$ [i]Proposed by kunny[/i]

2010 Today's Calculation Of Integral, 667
Tags: calculus , integration , geometry , geometric transformation , rotation , trigonometry , logarithms
Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

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

2007 Princeton University Math Competition, 10
Tags: logarithms
Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

2007 AMC 12/AHSME, 23
Tags: logarithms , geometry , quadratics
Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$ $ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$

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

2006 China Second Round Olympiad, 2
Tags: logarithms
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 $

2012 Today's Calculation Of Integral, 820
Tags: calculus , integration , analytic geometry , logarithms , limit , calculus computations
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

1950 AMC 12/AHSME, 25
Tags: logarithms
The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to: $\textbf{(A)}\ 725 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 3125 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None of these}$

1986 Vietnam National Olympiad, 3
Tags: algebra , polynomial , logarithms , algebra unsolved
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.

PEN M Problems, 15
Tags: logarithms , modular arithmetic , Recursive Sequences
For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of $f_{1991}(2^{1990})$.