Olympiad problems

2014 Harvard-MIT Mathematics Tournament, 17

Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$. (b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?

2009 Today's Calculation Of Integral, 452

Tags: logarithm , calculus , calculus computations , integration

Let $ a,\ b$ are postive constant numbers. (1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$ (2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$

2010 Contests, 522

Tags: geometry , function , calculus , limit , derivative , integration , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

1967 AMC 12/AHSME, 26

Tags: logarithm

If one uses only the tabular information $10^3=1000$, $10^4=10,000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, then the strongest statement one can make for $\log_{10}{2}$ is that it lies between: $\textbf{(A)}\ \frac{3}{10} \; \text{and} \; \frac{4}{11}\qquad \textbf{(B)}\ \frac{3}{10} \; \text{and} \; \frac{4}{12}\qquad \textbf{(C)}\ \frac{3}{10} \; \text{and} \; \frac{4}{13}\qquad \textbf{(D)}\ \frac{3}{10} \; \text{and} \; \frac{40}{132}\qquad \textbf{(E)}\ \frac{3}{11} \; \text{and} \; \frac{40}{132}$

2004 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , logarithm

For $x>0$, let $f(x)=x^x$. Find all values of $x$ for which $f(x)=f'(x)$.

1982 AMC 12/AHSME, 13

Tags: logarithm

If $a>1$, $b>1$, and $p=\frac{\log_b(\log_ba)}{\log_ba}$, then $a^n$ equals $\textbf {(A) } 1 \qquad \textbf {(B) } b \qquad \textbf {(C) } \log_ab \qquad \textbf {(D) } \log_ba \qquad \textbf {(E) } a^{\log_ba}$

2006 Putnam, A2

Tags: polynomial , function , probability , factorial , logarithm , algebra

Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

2003 National High School Mathematics League, 10

Tags: logarithm

$a,b,c,d$ are positive integers, and $\log_{a}b=\frac{3}{2},\log_{c}d=\frac{5}{4}$. If $a-c=9$, then $b-d=$________.

2011 Today's Calculation Of Integral, 751

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

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

2008 AMC 12/AHSME, 14

Tags: logarithm

A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$? $ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$

2006 China Second Round Olympiad, 2

Tags: logarithm

Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is ${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\ x>\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $

2013 ELMO Shortlist, 2

Tags: combinatorics , logarithm , induction , strong induction

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

1999 USAMTS Problems, 2

Tags: logarithm

Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Which is larger: $x_{42}$ or $x_{43}$?

2005 Brazil Undergrad MO, 5

Tags: calculus , function , induction , real analysis , integration , logarithm

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

1995 AMC 12/AHSME, 24

Tags: logarithm

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 $A+B+C$? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2005 France Team Selection Test, 4

Tags: inequalities , number theory , induction , floor function , logarithm , ceiling function

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2005 Today's Calculation Of Integral, 8

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

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

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2010 Today's Calculation Of Integral, 562

Tags: limit , inequalities , integration , calculus , logarithm , calculus computations

(1) Show the following inequality for every natural number $ k$. \[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\] (2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$. \[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]

2009 Today's Calculation Of Integral, 457

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

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2007 Moldova National Olympiad, 12.3

Tags: inequalities , logarithm , function , algebra

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: algebra , logarithm

Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

2010 Today's Calculation Of Integral, 626

Tags: limit , calculus , geometry , integration , logarithm , calculus computations

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

2008 China Western Mathematical Olympiad, 4

Tags: induction , algebra , logarithm , ceiling function

Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 \equal{} b$ and $ x_{n \plus{} 1} \equal{} ax^{m}_{n} \plus{} b$, $ n \equal{} 1,2,...$. Prove that (1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq \minus{} 2$; (2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq\frac {(m \minus{} 1)^{m \minus{} 1}}{m^m}$.