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

2001 Romania National Olympiad, 4
Tags: function , limit , integration , logarithm , real analysis
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: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

Today's calculation of integrals, 848
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2012 Today's Calculation Of Integral, 792
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

1993 National High School Mathematics League, 11
Tags: logarithm
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________.

2006 IMC, 5
Tags: inequalities , function , princeton , college , quadratic , integration , logarithm
Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

1949-56 Chisinau City MO, 51
Tags: algebra , logarithm , trigonometry
Determine graphically the number of roots of the equation $\sin x = \lg x$.

2013 NIMO Problems, 8
Tags: ceiling function , probability , number theory , relatively prime , logarithm
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

2005 Harvard-MIT Mathematics Tournament, 5
Tags: calculus , limit , logarithm
Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]

2009 Today's Calculation Of Integral, 460
Tags: calculus , integration , trigonometry , calculus computations , logarithm
$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.

2012 Today's Calculation Of Integral, 808
Tags: calculus , integration , limit , calculus computations , logarithm
For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\lim_{n\to\infty} a_n$.

1974 Canada National Olympiad, 1
Tags: induction , logarithm
i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

2007 Today's Calculation Of Integral, 237
Tags: calculus , integration , absolute value , calculus computations , logarithm
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

2012 Today's Calculation Of Integral, 773
Tags: calculus , integration , calculus computations , logarithm
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

2007 Moldova National Olympiad, 12.7
Tags: limit , integration , calculus , calculus computations , logarithm
Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

2011 Today's Calculation Of Integral, 688
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]

1965 AMC 12/AHSME, 31
Tags: logarithm
The number of real values of $ x$ satisfying the equality $ (\log_2x)(\log_bx) \equal{} \log_ab$, where $ a > 0$, $ b > 0$, $ a \neq 1$, $ b \neq 1$, is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite integer greater than 2} \qquad \textbf{(E)}\ \text{not finite}$

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

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2
Tags: calculus , integration , limit , logarithm , real analysis
For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2003 IMC, 2
Tags: calculus , integration , limit , trigonometry , inequalities , real analysis , logarithm
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

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

2004 Baltic Way, 8
Tags: algebra , polynomial , inequalities , function , number theory , prime number , logarithm
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2017 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , algebra , inequalities , logarithm
In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers

2011 Today's Calculation Of Integral, 734
Tags: calculus , integration , calculus computations , logarithm
Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

1967 AMC 12/AHSME, 4
Tags: logarithm
Given $\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}$, all logarithms to the same base and $x \not= 1$. If $\frac{b^2}{ac}=x^y$, then $y$ is: $ \text{(A)}\ \frac{q^2}{p+r}\qquad\text{(B)}\ \frac{p+r}{2q}\qquad\text{(C)}\ 2q-p-r\qquad\text{(D)}\ 2q-pr\qquad\text{(E)}\ q^2-pr$

2009 China Team Selection Test, 1
Tags: ceiling function , ratio , function , floor function , combinatorics , logarithm
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$