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

1959 AMC 12/AHSME, 23
Tags: algebra , logarithm
The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

2013 Today's Calculation Of Integral, 882
Tags: logarithm , limit , riemann sum , integration , calculus , calculus computations
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2001 Romania National Olympiad, 4
Tags: logarithm , real analysis , limit , integration , function
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2007 Today's Calculation Of Integral, 222
Tags: limit , calculus , logarithm , calculus computations , integration
Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

2005 Today's Calculation Of Integral, 48
Tags: trigonometry , integration , logarithm , limit , calculus , calculus computations
Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

2007 Today's Calculation Of Integral, 221
Tags: calculus computations , integration , logarithm , calculus
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.

1973 Miklós Schweitzer, 4
Tags: limit , number theory , logarithm
Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\] [i]P. Erdos[/i]

1991 Arnold's Trivium, 65
Tags: function , integration , geometry , logarithm , domain , 3d geometry , algebra
Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

1973 Miklós Schweitzer, 9
Tags: logarithm , probability and stats
Determine the value of \[ \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi],\] where $ \xi$ is a random variable and $ E$ denotes expectation. [i]Z. Daroczy[/i]

2025 District Olympiad, P1
Tags: logarithm
Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$ [i]Mathematical Gazette[/i]

2008 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: algebra , circumcircle , logarithm , inequalities
If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]

2007 Today's Calculation Of Integral, 230
Tags: calculus , logarithm , calculus computations , integration , function
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$.

2006 AMC 12/AHSME, 21
Tags: inequalities , geometry , ratio , logarithm , function
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$

1970 AMC 12/AHSME, 8
Tags: logarithm
If $a=\log_8225$ and $b=\log_215$, then $\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$

2003 Brazil National Olympiad, 3
Tags: combinatorics , logarithm
A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.

2011 Today's Calculation Of Integral, 748
Tags: integration , calculus , logarithm , trigonometry , calculus computations
Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

2014 AMC 12/AHSME, 18
Tags: logarithm , domain , number theory , function , algebra
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

2011 Kosovo National Mathematical Olympiad, 3
Tags: logarithm , inequalities
Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

2013 Today's Calculation Of Integral, 893
Tags: function , logarithm , derivative , integration , calculus , trigonometry , calculus computations
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2006 MOP Homework, 1
Tags: logarithm , ceiling function , combinatorics
Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

1957 AMC 12/AHSME, 5
Tags: logarithm
Through the use of theorems on logarithms \[ \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}}\] can be reduced to: $ \textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ 0\qquad \textbf{(E)}\ \log{\frac{a^2y}{d^2x}}$

2013 AIME Problems, 2
Tags: algebra , logarithm
Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

2009 Putnam, A2
Tags: integration , function , derivative , trigonometry , logarithm , calculus
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

2010 Today's Calculation Of Integral, 595
Tags: integration , trigonometry , logarithm , calculus , calculus computations
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

2011 Kazakhstan National Olympiad, 1
Tags: logarithm , algebra
Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $