Olympiad problems

2005 Today's Calculation Of Integral, 7

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

2013 NIMO Problems, 8

Tags: relatively prime , probability , logarithm , number theory , ceiling function

A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

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$

1968 AMC 12/AHSME, 23

Tags: logarithm

If all the logarithms are real numbers, the equality \[ \log(x+3)+\log (x-1) = \log (x^2-2x-3)\] is satisfied for: $\textbf{(A)}\ \text{all real values of}\ x \\ \qquad\textbf{(B)}\ \text{no real values of}\ x \\ \qquad\textbf{(C)}\ \text{all real values of}\ x\ \text{except}\ x=0 \\ \qquad\textbf{(D)}\ \text{no real values of}\ x\ \text{except}\ x=0 \\ \qquad\textbf{(E)}\ \text{all real values of}\ x\ \text{except}\ x=1$

2009 Today's Calculation Of Integral, 459

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

Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

2000 All-Russian Olympiad, 2

Tags: modular arithmetic , number theory , logarithm , ceiling function

Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

2011 Today's Calculation Of Integral, 723

Tags: calculus , integration , logarithm , calculus computations

Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$

2003 AMC 12-AHSME, 17

Tags: logarithm , function

If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$? $ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ 1$

2009 Putnam, A5

Tags: abstract algebra , group theory , logarithm

Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$

2009 China Second Round Olympiad, 2

Tags: logarithm , inequalities

Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

2005 Today's Calculation Of Integral, 1

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

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

1999 Federal Competition For Advanced Students, Part 2, 1

Tags: floor function , logarithm , number theory

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

2011 Today's Calculation Of Integral, 691

Tags: geometry , integration , calculus , geometric transformation , rotation , calculus computations , logarithm

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

2010 Today's Calculation Of Integral, 616

Tags: logarithm , integration , calculus computations , calculus

Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

2014 Contests, 1

Tags: induction , limit , logarithm

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

2008 Harvard-MIT Mathematics Tournament, 8

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

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

2008 Vietnam National Olympiad, 1

Tags: calculus , limit , logarithm , algebra

Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$

2007 AMC 12/AHSME, 17

Tags: logarithm

If $ a$ is a nonzero integer and $ b$ is a positive number such that $ ab^{2} \equal{} \log_{10}b,$ what is the median of the set $ \{0,1,a,b,1/b\}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ a \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \frac {1}{b}$

2010 Today's Calculation Of Integral, 538

Tags: integration , logarithm , calculus , calculus computations

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

2013 Waseda University Entrance Examination, 3

Tags: calculus , function , logarithm , calculus computations

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

2006 Harvard-MIT Mathematics Tournament, 5

Tags: logarithm , calculus , integration

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

2012 ELMO Shortlist, 6

Tags: least common multiple , induction , function , logarithm , number theory , pigeonhole principle , inequalities

Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$. [i]Linus Hamilton.[/i]

2019 Malaysia National Olympiad, 1

Tags: logarithm , algebra

Evaluate the following sum $$\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}$$

2009 ISI B.Stat Entrance Exam, 6

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

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

2011 Tokyo Instutute Of Technology Entrance Examination, 2

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

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]