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

2013 ELMO Shortlist, 5
Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

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]

2010 Today's Calculation Of Integral, 555
Tags: calculus , integration , derivative , function , calculus computations , logarithm
For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

1953 Putnam, A4
Tags: trigonometry , logarithm
From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

2010 Today's Calculation Of Integral, 560
Tags: calculus , integration , function , geometry , geometric transformation , rotation , logarithm
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$. (1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis. (2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

1967 AMC 12/AHSME, 23
Tags: limit , logarithm
If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{no finite number}$

2010 Today's Calculation Of Integral, 573
Tags: calculus , integration , geometry , trigonometry , calculus computations , logarithm
Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

2009 AMC 12/AHSME, 24
Tags: function , inequalities , logarithm
The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that \[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}} \]is defined? $ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

2014 Albania Round 2, 4
Tags: number theory , algebra , trigonometry , logarithm
Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

2009 Today's Calculation Of Integral, 482
Tags: calculus , integration , limit , geometry , rectangle , inequalities , logarithm
Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$

2010 Malaysia National Olympiad, 8
Tags: logarithm , inequalities
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.

2003 Brazil National Olympiad, 2
Tags: floor function , combinatorics , logarithm
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.

2000 Stanford Mathematics Tournament, 15
Tags: logarithm
Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?

2007 Romania Team Selection Test, 1
Tags: induction , calculus , derivative , logarithm , inequalities
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

1984 IMO Longlists, 60
Tags: inequalities , algebra , logarithm
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

Today's calculation of integrals, 882
Tags: calculus , integration , limit , calculus computations , riemann sum , logarithm
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2000 Croatia National Olympiad, Problem 4
Tags: algebra , floor function , logarithm
If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

2009 Today's Calculation Of Integral, 462
Tags: calculus , integration , trigonometry , calculus computations , logarithm
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$.

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}$

1964 AMC 12/AHSME, 1
Tags: logarithm
What is the value of $[\log_{10}(5\log_{10}100)]^2$? ${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $

2009 Harvard-MIT Mathematics Tournament, 2
Tags: hmmt , complex number , logarithm
Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

1996 AIME Problems, 2
Tags: floor function , logarithm
For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.

2005 Today's Calculation Of Integral, 15
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \frac{(x^2-1)^2}{x^4}dx$ [2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$ [3] $\int \sin 2x\cos 3xdx$ [4] $\int x\ln (x+1)dx$ [5] $\int \frac{x}{(x+3)^2}dx$

2010 Contests, 1
Tags: function , prime number , arithmetic sequence , number theory , logarithm
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

1992 Brazil National Olympiad, 2
Tags: number theory , logarithm
Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.