Olympiad problems

2010 Today's Calculation Of Integral, 540

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

1995 Irish Math Olympiad, 1

Tags: inequalities , induction , logarithms , inequalities proposed

Prove that for every positive integer $ n$, $ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$

2009 Today's Calculation Of Integral, 518

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

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

2012 Today's Calculation Of Integral, 812

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

Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$

Today's calculation of integrals, 894

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

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

1959 AMC 12/AHSME, 23

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

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

2019 CMI B.Sc. Entrance Exam, 6

Tags: logarithms , definite integrals , lebinitz rule , calculus

$(a)$ Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*} $(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\ \\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\ \\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

1953 AMC 12/AHSME, 39

Tags: logarithms

The product, $ \log_a b \cdot \log_b a$ is equal to: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ a \qquad\textbf{(C)}\ b \qquad\textbf{(D)}\ ab \qquad\textbf{(E)}\ \text{none of these}$

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

1998 South africa National Olympiad, 1

Tags: logarithms , number theory , prime factorization , rational number , irrational number

Is $\log_{10}{8}$ rational?

2006 Pre-Preparation Course Examination, 7

Tags: logarithms , function , limit , combinatorics proposed , combinatorics

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

2005 Today's Calculation Of Integral, 4

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

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

1991 AMC 12/AHSME, 20

Tags: geometry , algebra , polynomial , Vieta , logarithms , AMC

The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

2006 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , integration , logarithms

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

PEN E Problems, 15

Tags: logarithms , inequalities , number theory

Show that there exist two consecutive squares such that there are at least $1000$ primes between them.

2011 Today's Calculation Of Integral, 725

Tags: calculus , integration , logarithms , calculus computations

For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$

2011 China Second Round Olympiad, 9

Tags: logarithms , algebra unsolved , algebra

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

PEN G Problems, 17

Tags: inequalities , logarithms , Irrational numbers

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.

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$.

2010 Putnam, B5

Tags: Putnam , function , limit , integration , induction , inequalities , logarithms

Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

2014 AMC 10, 25

Tags: logarithms , inequalities , AMC , AIME , function , floor function

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

2007 Today's Calculation Of Integral, 225

Tags: calculus , integration , geometry , logarithms , ratio , inequalities , function

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.