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.

AND:
OR:
NO:

Found problems: 913

2005 AMC 12/AHSME, 23
Tags: parabola , logarithm , conic
Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

2005 Brazil Undergrad MO, 5
Tags: real analysis , calculus , integration , function , induction , logarithm
Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

2004 AIME Problems, 12
Tags: floor function , inequalities , rectangle , number theory , relatively prime , logarithm , geometry
Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

2011 Today's Calculation Of Integral, 753
Tags: calculus , integration , limit , calculus computations , logarithm
Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

2005 Today's Calculation Of Integral, 2
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$ [2]$\int \frac{dx}{\cos ^2 (3x+4)}$ [3]$\int (x-1)\sqrt[3]{x-2}dx$ [4]$\int x\cdot 3^{x^2+1}dx$ [5]$\int \frac{dx}{\sqrt{1-x}}dx$

2013 USA TSTST, 2
Tags: floor function , inequalities , function , algebra , logarithm
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.

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

2018 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: inequalities , algebra , logarithm
If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$

2011 Today's Calculation Of Integral, 724
Tags: calculus , integration , limit , calculus computations , logarithm
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2009 Today's Calculation Of Integral, 496
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4
Tags: parameterization , calculus , integration , logarithm , real analysis
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

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

2010 Today's Calculation Of Integral, 581
Tags: calculus , integration , geometry , inequalities , calculus computations , logarithm
For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.

2005 Today's Calculation Of Integral, 9
Tags: calculus , integration , trigonometry , absolute value , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

2007 Iran MO (3rd Round), 8
Tags: trigonometry , limit , ceiling function , factorial , floor function , logarithm
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

2010 Today's Calculation Of Integral, 616
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

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

1989 Canada National Olympiad, 3
Tags: modular arithmetic , algebra , binomial theorem , number theory , divisibility tests , logarithm
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

2019 CMI B.Sc. Entrance Exam, 6
Tags: lebinitz rule , calculus , definite integral , logarithm
$(a)$ Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*} $(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\ \\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\ \\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

1959 AMC 12/AHSME, 45
Tags: logarithm , algebra
If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals: $ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $

2010 Today's Calculation Of Integral, 575
Tags: calculus , integration , function , trigonometry , calculus computations , logarithm
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

2012 Postal Coaching, 3
Tags: function , floor function , algebra , integer part , logarithm
Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

2015 China National Olympiad, 1
Tags: number theory , logarithm
Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

2009 Vietnam Team Selection Test, 1
Tags: calculus , quadratic , function , logarithm , inequalities
Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2007 Romania National Olympiad, 2
Tags: algebra , logarithm
Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]