Olympiad problems

2014 USAMTS Problems, 3:

Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that: (i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$ (ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$ Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$

2023 AMC 12/AHSME, 6

Tags: AMC , AMC 12 , 2023 AMC , 2023 AMC 12A , logarithms , graph

Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? $\textbf{(A)}~2\sqrt{11}\qquad\textbf{(B)}~4\sqrt{3}\qquad\textbf{(C)}~8\qquad\textbf{(D)}~4\sqrt{5}\qquad\textbf{(E)}~9$

2009 Indonesia TST, 2

Tags: inequalities , logarithms , number theory , relatively prime , number theory proposed

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

2006 Moldova National Olympiad, 10.4

Tags: parameterization , logarithms , algebra unsolved , algebra

Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \] has a unique solution.

2008 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , function , logarithms , calculus computations

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

2013 ELMO Shortlist, 5

Tags: inequalities , logarithms , function , calculus , derivative , inequalities unsolved , 3-variable inequality

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]

2008 Federal Competition For Advanced Students, Part 2, 1

Tags: inequalities , function , logarithms , inequalities unsolved

Prove the inequality \[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3} \] holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.

2007 Tuymaada Olympiad, 4

Tags: probability , logarithms , limit , ratio , greatest common divisor , number theory , prime numbers

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

2019 AMC 12/AHSME, 15

Tags: logarithms , AMC , AMC 12 , AMC 12 A , 2019 AMC 12A , 2019 AMC

Positive real numbers $a$ and $b$ have the property that \[ \sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100 \] and all four terms on the left are positive integers, where $\text{log}$ denotes the base 10 logarithm. What is $ab$? $\textbf{(A) } 10^{52} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200} $

2007 Today's Calculation Of Integral, 175

Tags: calculus , integration , logarithms , calculus computations

Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: calculus , integration , limit , logarithms , real analysis , real analysis unsolved

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 $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

1992 Brazil National Olympiad, 2

Tags: logarithms , number theory proposed , number theory

Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.

2000 AMC 12/AHSME, 23

Tags: logarithms , probability

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$

1987 India National Olympiad, 2

Tags: logarithms , algebra proposed , algebra

Determine the largest number in the infinite sequence \[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]

2004 IMC, 5

Tags: integration , logarithms , inequalities , calculus , IMC , college contests

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

1996 Putnam, 2

Tags: Putnam , inequalities , induction , logarithms , geometry , rectangle , integration

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

1979 IMO Longlists, 37

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.

1994 Vietnam National Olympiad, 1

Tags: logarithms , algebra , system of equations , algebra proposed

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

2007 Princeton University Math Competition, 5

Tags: logarithms

Round to the nearest tenth: $\log_6 (6^2-6+1) + 3\log_6 (5) - \frac{1}{2}\log_6 (9)$.

1967 Miklós Schweitzer, 4

Tags: logarithms , real analysis , real analysis unsolved , Miklos Schweitzer , inequalities

Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that \[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\] [i]L. Leinder[/i]

1957 Putnam, A6

Tags: Putnam , limit , logarithms

Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$ \lim_{n \to \infty} S_n = a-1.$$

PEN J Problems, 11

Tags: function , modular arithmetic , inequalities , induction , floor function , logarithms , number theory

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

2002 District Olympiad, 3

Tags: number theory , logarithms , inequalities

Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $ Show that $ a<b. $

2011 Today's Calculation Of Integral, 686

Tags: calculus , integration , analytic geometry , geometry , perimeter , limit , logarithms

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]

2025 District Olympiad, P1

Tags: logarithms

Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$ [i]Mathematical Gazette[/i]