Olympiad problems

2005 Today's Calculation Of Integral, 17

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

2007 AIME Problems, 12

Tags: calculus , integration , logarithms , ratio , AMC , USA(J)MO , USAMO

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

2015 AMC 12/AHSME, 14

Tags: logarithms , AMC

What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$

2007 All-Russian Olympiad Regional Round, 10.2

Tags: inequalities , logarithms , inequalities proposed

Prove that the inequality $ (x^{k}\minus{}y^{k})^{n}<(x^{n}\minus{}y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.

2012 Today's Calculation Of Integral, 774

Tags: calculus , integration , logarithms , calculus computations

Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$

1994 APMO, 5

Tags: floor function , logarithms , number theory unsolved , number theory

You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

2016 AIME Problems, 3

Tags: AMC , AIME , AIME II , logarithms

Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x) &= 5 \\ \log_3(xyz-3+\log_5 y) &= 4 \\ \log_4(xyz-3+\log_5 z) &= 4. \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.

1995 AMC 12/AHSME, 24

Tags: logarithms

There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that \[A \log_{200} 5 + B \log_{200} 2 = C. \] What is $A+B+C$? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2009 Today's Calculation Of Integral, 447

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

Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

2006 ISI B.Stat Entrance Exam, 8

Tags: logarithms , limit , algebra proposed , algebra

Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that \[\frac{\log_2x}{x}=c\] has only one real solution.

2002 AMC 12/AHSME, 22

Tags: logarithms

For all integers $ n$ greater than $ 1$, define $ a_n \equal{} \frac {1}{\log_n 2002}$. Let $ b \equal{} a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5$ and $ c \equal{} a_{10} \plus{} a_{11} \plus{} a_{12} \plus{} a_{13} \plus{} a_{14}$. Then $ b \minus{} c$ equals $ \textbf{(A)}\ \minus{} 2 \qquad \textbf{(B)}\ \minus{} 1 \qquad \textbf{(C)}\ \frac {1}{2002} \qquad \textbf{(D)}\ \frac {1}{1001} \qquad \textbf{(E)}\ \frac {1}{2}$

2012 Today's Calculation Of Integral, 788

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

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

PEN O Problems, 40

Tags: floor function , induction , logarithms

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

2006 Iran Team Selection Test, 2

Tags: ceiling function , logarithms , algebra proposed , algebra

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

1957 AMC 12/AHSME, 28

Tags: logarithms

If $ a$ and $ b$ are positive and $ a\not\equal{} 1,\,b\not\equal{} 1$, then the value of $ b^{\log_b{a}}$ is: $ \textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad \textbf{(D)}\ \text{zero}\qquad \textbf{(E)}\ \text{one}$

1989 IMO Longlists, 45

Tags: logarithms , number theory , algebra unsolved , 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.

2008 Harvard-MIT Mathematics Tournament, 4

Tags: quadratics , limit , logarithms , calculus , calculus computations

([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.

2009 Vietnam National Olympiad, 4

Tags: algebra , polynomial , induction , calculus , integration , logarithms , function

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$.

1959 AMC 12/AHSME, 45

Tags: logarithms , AMC , logarithm , algebra , AMC 12

If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals: $ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $

1974 Canada National Olympiad, 1

Tags: logarithms , induction

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

2010 Today's Calculation Of Integral, 621

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

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]

1962 AMC 12/AHSME, 28

Tags: logarithms

The set of $ x$-values satisfying the equation $ x^{\log_{10} x} \equal{} \frac{x^3}{100}$ consists of: $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \text{10, only} \qquad \textbf{(C)}\ \text{100, only} \qquad \textbf{(D)}\ \text{10 or 100, only} \qquad \textbf{(E)}\ \text{more than two real numbers.}$

2014 Harvard-MIT Mathematics Tournament, 10

Tags: logarithms , function , calculus , derivative

Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.

2014 Cezar Ivănescu, 1

Tags: inequalities , algebra , logarithms

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $