Olympiad problems

2011 Today's Calculation Of Integral, 734

Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

2022 District Olympiad, P1

Tags: romania , logarithms , algebra

Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

1979 IMO Shortlist, 14

Tags: logarithms , IMO 1979 , IMO Shortlist

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

1978 IMO Longlists, 28

Tags: function , logarithms , algebra unsolved , algebra

Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy \[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\] Prove that: $(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$; $(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$). Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.

2012 Bogdan Stan, 3

Tags: limits , Floor , limits of sequences , inequalities , logarithms , Bernoulli s Inequality

$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $ [i]Taclit Daniela Nadia[/i]

2015 China National Olympiad, 1

Tags: logarithms , number theory proposed , number theory

Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

2005 Today's Calculation Of Integral, 18

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Calculate the following indefinite integrals. [1] $\int (\sin x+\cos x)^4 dx$ [2] $\int \frac{e^{2x}}{e^x+1}dx$ [3] $\int \sin ^ 4 xdx$ [4] $\int \sin 6x\cos 2xdx$ [5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$

1988 Irish Math Olympiad, 4

Tags: trigonometry , logarithms , geometry unsolved , geometry

Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $ a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $ log a^2 = log b^2 + log c^2 - log(2bc cos A) $ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

2009 Today's Calculation Of Integral, 471

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

1950 AMC 12/AHSME, 18

Tags: logarithms

Of the following (1) $ a(x\minus{}y)\equal{}ax\minus{}ay$ (2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$ (3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$ (4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$ (5) $ a(xy)\equal{}ax\times ay$ $\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\ \textbf{(E)}\ \text{Only 1 is true}$

1991 Arnold's Trivium, 65

Tags: function , logarithms , algebra , domain , integration , geometry , 3D geometry

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

2005 AMC 10, 17

Tags: logarithms

Suppose that $ 4^a \equal{} 5$, $ 5^b \equal{} 6$, $ 6^c \equal{} 7$, and $ 7^d \equal{} 8$. What is $ a\cdot b\cdot c\cdot d$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{5}{2}\qquad \textbf{(E)}\ 3$

1962 AMC 12/AHSME, 17

Tags: logarithms

If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is: $ \textbf{(A)}\ \frac{b}{2} \qquad \textbf{(B)}\ \frac{2b}{3}\qquad \textbf{(C)}\ b \qquad \textbf{(D)}\ \frac{3b}{2} \qquad \textbf{(E)}\ 2b$

2011 USA TSTST, 3

Tags: analytic geometry , logarithms , USA(J)MO , USAMO , number theory , relatively prime , USA TSTST

Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation \[ \sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2). \]

Oliforum Contest II 2009, 5

Tags: function , logarithms , combinatorics proposed , combinatorics

Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ \{a_i\}_{i \in \mathbb{N}}$ such that $ a_0 \equal{} 1$ e $ a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell$, where $ \ell \equal{} 0$ if $ \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0$, and $ \ell \equal{} 1$ otherwise. How many distinct elements are in the set $ S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}$? [i](Paolo Leonetti)[/i]

IV Soros Olympiad 1997 - 98 (Russia), 11.5

Tags: number theory , logarithms

Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.

2010 Today's Calculation Of Integral, 610

Tags: calculus , integration , logarithms , calculus computations

Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$ proposed by kunny

PEN A Problems, 107

Tags: logarithms , inequalities , rearrangement inequality , Divisibility Theory

Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.

1958 AMC 12/AHSME, 25

Tags: logarithms , AMC , AIME

If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals: $ \textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$

1953 AMC 12/AHSME, 47

Tags: inequalities , logarithms , calculus

If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

1991 IMTS, 1

Tags: floor function , logarithms , function

For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases: (i) $10 \leq n \leq 99$ (ii) $100 \leq n \leq 999$ (iii) $1000 \leq n \leq 9999$ (iv) $10000 \leq n \leq 99999$

2000 AIME Problems, 9

Tags: logarithms , AMC , AIME , algebra , system of equations , AIME I , binomial theorem

The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$

1976 Miklós Schweitzer, 11

Tags: logarithms , probability , geometry , geometric transformation , reflection , probability and stats

Let $ \xi_1,\xi_2,...$ be independent, identically distributed random variables with distribution \[ P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .\] Write $ S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,$ and \[ T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .\] Prove that $ \liminf_{n \rightarrow \infty} (\log n)T_n=0$ with probability one. [i]P. Revesz[/i]

2020 AMC 12/AHSME, 10

Tags: AMC , AMC 12 , AMC 12 A , logarithms , 2020 AMC , 2020 AMC 12A

There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\] What is the sum of the digits of $n?$ $\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

2005 Brazil Undergrad MO, 5

Tags: calculus , integration , logarithms , function , induction , real analysis , real analysis solved

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]