Olympiad problems

1961 AMC 12/AHSME, 6

Tags: logarithm

When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: ${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $

PEN E Problems, 14

Tags: algebra , function , number theory , polynomial , logarithm , limit , gauss

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

2014 Albania Round 2, 4

Tags: number theory , trigonometry , algebra , logarithm

Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

2011 Pre-Preparation Course Examination, 5

Tags: calculus , logarithm , function , inequalities , number theory , integration , prime number

suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent: [b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$ [b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).

2016 CMIMC, 2

Tags: logarithm , algebra

Suppose that some real number $x$ satisfies \[\log_2 x + \log_8 x + \log_{64} x = \log_x 2 + \log_x 16 + \log_x 128.\] Given that the value of $\log_2 x + \log_x 2$ can be expressed as $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are coprime positive integers and $b$ is squarefree, compute $abc$.

2009 China Team Selection Test, 1

Tags: function , floor function , combinatorics , logarithm , ceiling function , ratio

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.$

2010 Today's Calculation Of Integral, 555

Tags: function , logarithm , derivative , calculus , integration , calculus computations

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

1994 Vietnam National Olympiad, 1

Tags: algebra , system of equations , logarithm

Find all real solutions to \[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\] \[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\] \[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]

1997 All-Russian Olympiad Regional Round, 11.6

Tags: algebra , inequalities , logarithm

Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

2009 Today's Calculation Of Integral, 518

Tags: trigonometry , integration , logarithm , calculus , calculus computations

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

2024 AIME, 2

Tags: logarithm

Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?

2015 Mathematical Talent Reward Programme, SAQ: P 2

Tags: algebra , logarithm

Let $x, y$ be numbers in the interval (0,1) such that for some $a>0, a \neq 1$ $$\log _{x} a+\log _{y} a=4 \log _{x y} a$$Prove that $x=y$

1976 Euclid, 5

Tags: logarithm

Source: 1976 Euclid Part A Problem 5 ----- If $\log_8 m+\log_8 \frac{1}{6}=\frac{2}{3}$, then $m$ equals $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{23}{6} \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 24$