Olympiad problems

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

Tags: parameterization , calculus , integration , logarithms , real analysis , real analysis unsolved

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, 1

Tags: probability , calculus , integration , geometry , logarithms , trapezoid

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

2007 Romania National Olympiad, 2

Tags: logarithms , algebra unsolved , algebra

Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]

2012 ELMO Shortlist, 6

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

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]

2009 Today's Calculation Of Integral, 507

Tags: calculus , integration , logarithms , calculus computations

Evaluate \[ \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]

2012 Today's Calculation Of Integral, 773

Tags: calculus , integration , logarithms , calculus computations

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

2010 Today's Calculation Of Integral, 523

Tags: calculus , integration , logarithms , calculus computations

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2007 Purple Comet Problems, 12

Tags: logarithms

Find the maximum possible value of $8\cdot 27^{\log_6 x}+27\cdot 8^{\log_6 x}-x^3$ as $x$ varies over the positive real numbers.

2006 China Second Round Olympiad, 5

Tags: logarithms

Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$

2009 Today's Calculation Of Integral, 488

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

For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$

2009 Putnam, A5

Tags: Putnam , abstract algebra , group theory , logarithms , college contests

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

2011 Today's Calculation Of Integral, 763

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

Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$

2004 Romania National Olympiad, 2

Tags: function , algebra , polynomial , logarithms , calculus , integration , inequalities

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

2000 Baltic Way, 20

Tags: integration , trigonometry , logarithms , algebra proposed , algebra

For every positive integer $n$, let \[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\] Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.

2019 AMC 12/AHSME, 12

Tags: AMC , AMC 12 , AMC 12 A , logarithms , algebra , quadratic formula , 2019 AMC 12A

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? $\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$

1952 AMC 12/AHSME, 18

Tags: logarithms

$ \log p \plus{} \log q \equal{} \log (p \plus{} q)$ only if: $ \textbf{(A)}\ p \equal{} q \equal{} 0 \qquad\textbf{(B)}\ p \equal{} \frac {q^2}{1 \minus{} q} \qquad\textbf{(C)}\ p \equal{} q \equal{} 1$ $ \textbf{(D)}\ p \equal{} \frac {q}{q \minus{} 1} \qquad\textbf{(E)}\ p \equal{} \frac {q}{q \plus{} 1}$

1973 Miklós Schweitzer, 4

Tags: limit , logarithms , number theory proposed , number theory

Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\] [i]P. Erdos[/i]

2008 AMC 12/AHSME, 23

Tags: logarithms , floor function , geometry , 3D geometry , pyramid , symmetry , number theory

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

1982 AMC 12/AHSME, 13

Tags: logarithms

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

1964 AMC 12/AHSME, 1

Tags: logarithms , AMC

What is the value of $[\log_{10}(5\log_{10}100)]^2$? ${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $

2012 China Northern MO, 6

Tags: function , logarithms , inequalities proposed , inequalities

Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

2007 Today's Calculation Of Integral, 254

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$ Sorry, I have deleted my first post because that was wrong. kunny

Today's calculation of integrals, 889

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

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

2008 Junior Balkan Team Selection Tests - Romania, 1

Tags: induction , logarithms , modular arithmetic , number theory , number theory proposed

Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

2009 Today's Calculation Of Integral, 483

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

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$