Olympiad problems

1973 Miklós Schweitzer, 3

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

1967 Miklós Schweitzer, 4

Tags: logarithm , inequalities , real analysis

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]

2004 China Team Selection Test, 2

Tags: number theory , floor function , logarithm

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

2007 Today's Calculation Of Integral, 179

Tags: calculus , calculus computations , logarithm , integration

Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

Tags: logarithm , real analysis , calculus , parameterization , integration

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

2007 Princeton University Math Competition, 5

Tags: logarithm

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

2009 Today's Calculation Of Integral, 458

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

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

2013 Romania Team Selection Test, 1

Tags: combinatorics , logarithm

Fix a point $O$ in the plane and an integer $n\geq 3$. Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that: (a) No disc in $\mathcal{D}$ contains the point $O$; and (b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$. Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$.

2007 Today's Calculation Of Integral, 198

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

Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2010 Today's Calculation Of Integral, 637

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

For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

2003 AIME Problems, 4

Tags: trigonometry , logarithm

Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$.

1994 Turkey Team Selection Test, 2

Tags: algebra , limit , logarithm

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

1974 USAMO, 2

Tags: function , inequalities , logarithm

Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]

2010 Today's Calculation Of Integral, 584

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

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

1988 Irish Math Olympiad, 4

Tags: logarithm , trigonometry , 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?

1965 AMC 12/AHSME, 6

Tags: logarithm

If $ 10^{\log_{10}9} \equal{} 8x \plus{} 5$ then $ x$ equals: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {5}{8} \qquad \textbf{(D)}\ \frac {9}{8} \qquad \textbf{(E)}\ \frac {2\log_{10}3 \minus{} 5}{8}$

2011 AMC 12/AHSME, 17

Tags: logarithm

Let $f\left(x\right)=10^{10x}, g\left(x\right)=\log_{10}\left(\frac{x}{10}\right), h_1\left(x\right)=g\left(f\left(x\right)\right),$ and $h_n\left(x\right)=h_1\left(h_{n-1}\left(x\right)\right)$ for integers $n \ge 2$. What is the sum of the digits of $h_{2011}\left(1\right)$? $ \textbf{(A)}\ 16,081 \qquad \textbf{(B)}\ 16,089 \qquad \textbf{(C)}\ 18,089 \qquad \textbf{(D)}\ 18,098 \qquad \textbf{(E)}\ 18,099 $

1995 Baltic Way, 9

Tags: induction , algebra , logarithm

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

2010 Contests, 1

Tags: logarithm

Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$.

2014 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , derivative , logarithm

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 $L=1+\log_\phi(c)$.

1987 India National Olympiad, 2

Tags: algebra , logarithm

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

2011 Today's Calculation Of Integral, 743

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

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

2012 Postal Coaching, 3

Tags: algebra , function , floor function , integer part , logarithm

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

1950 AMC 12/AHSME, 26

Tags: logarithm

If $ \log_{10}{m} \equal{} b \minus{} \log_{10}{n}$, then $ m$= $\textbf{(A)}\ \dfrac{b}{n} \qquad \textbf{(B)}\ bn \qquad \textbf{(C)}\ 10^b n\qquad \textbf{(D)}\ b-10^n \qquad \textbf{(E)}\ \dfrac{10^b}{n}$

1959 AMC 12/AHSME, 45

Tags: algebra , logarithm

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 $