Olympiad problems

2005 AMC 12/AHSME, 23

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

2013 Today's Calculation Of Integral, 882

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

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2010 Putnam, A6

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

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

2005 AIME Problems, 5

Tags: floor function , logarithm

Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.

2014 Singapore Senior Math Olympiad, 3

Tags: logarithm

Find the value of $\frac{\log_59\log_75\log_37}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}}$ $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7 $

1975 AMC 12/AHSME, 19

Tags: logarithm

Which positive numbers $ x$ satisfy the equation $ (\log_3x)(\log_x5)\equal{}\log_35$? $ \textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad$ $ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are }$ $ \text{positive integers} \qquad$ $ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad \textbf{(E)}\ \text{none of these}$

2019 Jozsef Wildt International Math Competition, W. 6

Tags: integration , trigonometry , calculus , logarithm

Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$

1966 IMO Shortlist, 30

Tags: inequalities , calculus , logarithm

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2006 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , integration , logarithm

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

2002 Moldova National Olympiad, 2

Tags: inequalities , logarithm

Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality: $ \log_c\log_c b \plus{} \log_b\log_b a \plus{} \log_a\log_a c\geq 0$

1950 AMC 12/AHSME, 25

Tags: logarithm

The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to: $\textbf{(A)}\ 725 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 3125 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None of these}$

2006 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , integration , logarithm

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

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 $\textbf {(A) } 1 \qquad \textbf {(B) } b \qquad \textbf {(C) } \log_ab \qquad \textbf {(D) } \log_ba \qquad \textbf {(E) } a^{\log_ba}$

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

1987 IMO Longlists, 17

Tags: algebra , logarithm

Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.

2007 China Northern MO, 2

Tags: function , algebra , logarithm

Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $ f(x) = 0$. b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]

PEN G Problems, 17

Tags: inequalities , logarithm , irrational number

Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.

2003 China Team Selection Test, 1

Tags: function , induction , logarithm , algebra

Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

2011 Today's Calculation Of Integral, 727

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

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2018 India PRMO, 14

Tags: trigonometry , logarithm

If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?

1953 AMC 12/AHSME, 21

Tags: logarithm

If $ \log_{10} (x^2\minus{}3x\plus{}6)\equal{}1$, the value of $ x$ is: $ \textbf{(A)}\ 10\text{ or }2 \qquad\textbf{(B)}\ 4\text{ or }\minus{}2 \qquad\textbf{(C)}\ 3\text{ or }\minus{}1 \qquad\textbf{(D)}\ 4\text{ or }\minus{}1\\ \textbf{(E)}\ \text{none of these}$

2012 Waseda University Entrance Examination, 4

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

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 $Q$.

2007 Princeton University Math Competition, 5

Tags: logarithm

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

2005 Today's Calculation Of Integral, 3

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

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