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

2010 Today's Calculation Of Integral, 621
Tags: calculus , integration , limit , real analysis , calculus computations , logarithm
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

2012 Online Math Open Problems, 26
Tags: modular arithmetic , floor function , number theory , least common multiple , algebra , logarithm
Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

2012 Today's Calculation Of Integral, 804
Tags: calculus , integration , derivative , function , calculus computations , logarithm
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

1966 AMC 12/AHSME, 9
Tags: logarithm
If $x=(\log_82)^{(\log_28)}$, then $\log_3x$ equals: $\text{(A)} \ -3 \qquad \text{(B)} \ -\frac13 \qquad \text{(C)} \ \frac13 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 9$

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

1984 AMC 12/AHSME, 14
Tags: logarithm
The product of all real roots of the equation $x^{\log_{10} x} = 10$ is A. 1 B. -1 C. 10 D. $10^{-1}$ E. None of these

2008 AMC 12/AHSME, 14
Tags: logarithm
A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$? $ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$

2000 AMC 12/AHSME, 23
Tags: probability , logarithm
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$

2011 Tokyo Instutute Of Technology Entrance Examination, 2
Tags: calculus , integration , trigonometry , calculus computations , logarithm
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

2005 Today's Calculation Of Integral, 42
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

2023 China Second Round, 2
Tags: algebra , logarithm
if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$

1963 AMC 12/AHSME, 5
Tags: logarithm
If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then: $\textbf{(A)}\ x<0 \qquad \textbf{(B)}\ -1<x<1 \qquad \textbf{(C)}\ 0<x\le 1 $ $ \textbf{(D)}\ -1<x<0 \qquad \textbf{(E)}\ 0<x<1$

2010 Today's Calculation Of Integral, 631
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

1972 AMC 12/AHSME, 29
Tags: function , logarithm
If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is $\textbf{(A) }-f(x)\qquad\textbf{(B) }2f(x)\qquad\textbf{(C) }3f(x)\qquad$ $\textbf{(D) }\left[f(x)\right]^2\qquad \textbf{(E) }[f(x)]^3-f(x)$

2010 Today's Calculation Of Integral, 537
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

2007 Today's Calculation Of Integral, 184
Tags: calculus , integration , function , inequalities , real analysis , calculus computations , logarithm
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

1953 AMC 12/AHSME, 47
Tags: inequalities , calculus , logarithm
If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2011 China Second Round Olympiad, 3
Tags: inequalities , algebra , logarithm
Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$.

2012 Today's Calculation Of Integral, 788
Tags: calculus , integration , function , analytic geometry , calculus computations , logarithm
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

2010 Putnam, A4
Tags: modular arithmetic , number theory , greatest common divisor , putnam number theory , logarithm
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

1984 All Soviet Union Mathematical Olympiad, 392
Tags: compare , algebra , logarithm
What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).

2008 Putnam, B2
Tags: integration , limit , calculus , factorial , logarithm
Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

1969 AMC 12/AHSME, 25
Tags: inequalities , absolute value , logarithm
If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is: $\textbf{(A) }2\sqrt6\qquad \textbf{(B) }6\qquad \textbf{(C) }8\sqrt2\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{none of these.}$

2009 Today's Calculation Of Integral, 516
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2006 AIME Problems, 9
Tags: ratio , modular arithmetic , logarithm
The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$.