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

1997 AMC 12/AHSME, 21
Tags: function , matrix , logarithm , integration , linear algebra , calculus
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$? $ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$

2009 Vietnam National Olympiad, 4
Tags: function , polynomial , logarithm , integration , induction , calculus , algebra
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.

2014 Singapore Senior Math Olympiad, 11
Tags: logarithm
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$

2019 Jozsef Wildt International Math Competition, W. 4
Tags: logarithm , inequalities
If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$

2010 Today's Calculation Of Integral, 631
Tags: calculus , logarithm , integration , calculus computations
Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

2009 Today's Calculation Of Integral, 446
Tags: calculus computations , calculus , logarithm , integration
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2009 Indonesia TST, 1
Tags: logarithm , combinatorics , inequalities , induction
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

1985 Canada National Olympiad, 4
Tags: prime factorization , logarithm , inequalities , number theory , floor function
Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

V Soros Olympiad 1998 - 99 (Russia), 11.1
Tags: algebra , logarithm , trigonometry
Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.

1984 IMO Shortlist, 20
Tags: logarithm , inequalities , algebra
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2003 Estonia National Olympiad, 2
Tags: logarithm , equation , algebra
Solve the equation $\sqrt{x} = \log_2 x$.

1961 AMC 12/AHSME, 30
Tags: logarithm
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}} $

2013 AIME Problems, 8
Tags: logarithm , domain , function , modular arithmetic , algebra , trigonometry
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

2011 Albania National Olympiad, 1
Tags: logarithm , function , graphing lines , algebra , analytic geometry , slope
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

2012 Today's Calculation Of Integral, 774
Tags: logarithm , calculus , calculus computations , integration
Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$

1971 AMC 12/AHSME, 21
Tags: logarithm
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to $\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$

1999 National High School Mathematics League, 3
Tags: logarithm
If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

2011 Today's Calculation Of Integral, 694
Tags: quadratic , logarithm , integration , inequalities , calculus computations , calculus
Prove the following inequality: \[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\] created by kunny

2011 Today's Calculation Of Integral, 686
Tags: perimeter , logarithm , integration , geometry , analytic geometry , calculus , limit
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

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

1989 IMO Longlists, 45
Tags: number theory , logarithm , algebra
Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.

2013 Today's Calculation Of Integral, 896
Tags: logarithm , integration , function , calculus computations , calculus , limit
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

1991 Romania Team Selection Test, 3
Tags: logarithm , induction , combinatorics
Prove the following identity for every $ n\in N$: $ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$

2012 China Team Selection Test, 1
Tags: logarithm , function , number theory , floor function
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2011 Today's Calculation Of Integral, 725
Tags: calculus computations , calculus , logarithm , integration
For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$