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

1993 National High School Mathematics League, 11
Tags: logarithms
Four real numbers $x_0>x_1>x_2>x_3>0$, if $\log_{\frac{x_0}{x_1}}1993+\log_{\frac{x_1}{x_2}}1993+\log_{\frac{x_2}{x_3}}1993\geq k\cdot\log_{\frac{x_0}{x_3}}1993$ for all $x_0,x_1,x_2,x_3$, then the maximum value of $k$ is________.

2010 Today's Calculation Of Integral, 595
Tags: calculus , integration , trigonometry , logarithms , 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

2024 AMC 12/AHSME, 15
Tags: geometry , AMC , AMC 12 , 2024 AMC , 2024 AMC 12B , coordinate geometry , logarithms
A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

2000 All-Russian Olympiad, 2
Tags: ceiling function , logarithms , modular arithmetic , number theory unsolved , number theory
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.

2014 Miklós Schweitzer, 1
Tags: ceiling function , logarithms , combinatorics unsolved , combinatorics
Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

2009 ISI B.Stat Entrance Exam, 6
Tags: calculus , derivative , function , logarithms , integration , calculus computations
Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

2002 Putnam, 3
Tags: Putnam , inequalities , logarithms , college contests
Show that for all integers $n>1$, \[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]

1986 AIME Problems, 8
Tags: logarithms
Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$?

2012 Today's Calculation Of Integral, 799
Tags: calculus , integration , limit , logarithms , induction , strong induction , calculus computations
Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

1984 AMC 12/AHSME, 14
Tags: logarithms , AMC , AMC 12
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

2004 India IMO Training Camp, 1
Tags: inequalities , logarithms , function , inequalities unsolved
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

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

2009 Today's Calculation Of Integral, 485
Tags: calculus , integration , geometry , inequalities , function , logarithms , trigonometry
In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

2005 Today's Calculation Of Integral, 43
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate \[\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx\]

2010 Malaysia National Olympiad, 8
Tags: logarithms , inequalities , inequalities unsolved
Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.

2011 Today's Calculation Of Integral, 723
Tags: calculus , integration , logarithms , calculus computations
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$

2005 China Team Selection Test, 2
Tags: logarithms , algebra unsolved , algebra
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2011 Today's Calculation Of Integral, 729
Tags: calculus , integration , logarithms , calculus computations
Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$

1984 AIME Problems, 5
Tags: logarithms , symmetry
Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.

2020 Kosovo National Mathematical Olympiad, 3
Tags: algebra , minimum value , logarithms , national olympiad , Olympiad , Kosovo
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

2019 Purple Comet Problems, 9
Tags: logarithms , algebra
Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$

2010 Today's Calculation Of Integral, 646
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2006 Romania Team Selection Test, 3
Tags: floor function , logarithms , function , algebra proposed , algebra
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 \[ \lfloor 25 x_{625} \rfloor = 625 . \]

2016 Korea USCM, 1
Tags: limit , real analysis , college contests , Riemann sum , logarithms
Find the following limit. \[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]

2009 Today's Calculation Of Integral, 457
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$