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

2014 JBMO Shortlist, 3
Tags: floor function , modular arithmetic , limit , logarithms , combinatorics solved , combinatorics
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2013 Today's Calculation Of Integral, 898
Tags: calculus , integration , logarithms , calculus computations
Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2013 Today's Calculation Of Integral, 889
Tags: calculus , integration , geometry , logarithms , calculus computations
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

2009 Today's Calculation Of Integral, 400
Tags: calculus , integration , function , logarithms , calculus computations
(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

1981 AMC 12/AHSME, 15
Tags: logarithms
If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$

2014 Harvard-MIT Mathematics Tournament, 3
Tags: HMMT , logarithms
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

2009 Today's Calculation Of Integral, 458
Tags: calculus , integration , geometry , logarithms , vector , calculus computations
Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

2012 Today's Calculation Of Integral, 803
Tags: calculus , integration , limit , logarithms , factorial , calculus computations
Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2009 AIME Problems, 11
Tags: logarithms , AMC
For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m \minus{} \log k| < \log n$. Find the sum of all possible values of the product $ mn$.

1961 AMC 12/AHSME, 30
Tags: logarithms
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ ${{ \textbf{(A)}\ \frac{a+b}{a+1} \qquad\textbf{(B)}\ \frac{2a+b}{a+1} \qquad\textbf{(C)}\ \frac{a+2b}{1+a} \qquad\textbf{(D)}\ \frac{2a+b}{1-a} }\qquad\textbf{(E)}\ \frac{a+2b}{1-a}} $

2009 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , integration , logarithms
Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2009 Today's Calculation Of Integral, 462
Tags: calculus , integration , trigonometry , logarithms , calculus computations
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.

1987 IMO Longlists, 17
Tags: logarithms , algebra unsolved , algebra
Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.

2011 IMC, 3
Tags: logarithms , limit , IMC , college contests
Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.

1985 AMC 12/AHSME, 15
Tags: logarithms
If $ a$ and $ b$ are positive numbers such that $ a^b \equal{} b^a$ and $ b \equal{} 9a$, then the value of $ a$ is: $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \sqrt [9] {9}\qquad \textbf{(D)}\ \sqrt [3] {9}\qquad \textbf{(E)}\ \sqrt [4] {3}$

2014 USAMO, 6
Tags: USA(J)MO , USAMO , inequalities , logarithms , Putnam
Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]

2022 VTRMC, 4
Tags: VTRMC , college contests , logarithms , series , Summation , infinite sum
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.

2007 Today's Calculation Of Integral, 170
Tags: calculus , integration , logarithms , algebra , partial fractions , calculus computations
Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2009 China Second Round Olympiad, 2
Tags: logarithms , inequalities proposed , inequalities
Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

2010 Today's Calculation Of Integral, 592
Tags: calculus , integration , logarithms , trigonometry , calculus computations
Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

2010 Romania National Olympiad, 4
Tags: logarithms , limit , real analysis , real analysis unsolved
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

2007 Today's Calculation Of Integral, 210
Tags: calculus , integration , logarithms , trigonometry , function , calculus computations
Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.

2010 Today's Calculation Of Integral, 547
Tags: calculus , integration , logarithms , function , derivative , real analysis , calculus computations
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2016 District Olympiad, 1
Tags: equation , algebra , logarithms
Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$

1998 AMC 12/AHSME, 12
Tags: logarithms , number theory , prime numbers
How many different prime numbers are factors of $ N$ if \[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11? \]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$