Olympiad problems

2011 Today's Calculation Of Integral, 753

Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

1965 AMC 12/AHSME, 6

Tags: logarithms

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 Today's Calculation Of Integral, 736

Tags: calculus , integration , logarithms , derivative , calculus computations

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

2019 AMC 12/AHSME, 23

Tags: AMC , AMC 12 , AMC 12 A , logarithms , 2019 AMC 12A , 2019 AMC

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}$$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and $$a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}$$ for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$? $\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

PEN J Problems, 8

Tags: logarithms , Divisor Functions

Prove that for any $ \delta\in[0,1]$ and any $ \varepsilon>0$, there is an $ n\in\mathbb{N}$ such that $ \left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon$.

Today's calculation of integrals, 892

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

Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$

1963 AMC 12/AHSME, 5

Tags: logarithms , AMC

If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then: $\textbf{(A)}\ x<0 \qquad \textbf{(B)}\ -1<x<1 \qquad \textbf{(C)}\ 0<x\le 1 $ $ \textbf{(D)}\ -1<x<0 \qquad \textbf{(E)}\ 0<x<1$

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: logarithms , algebra , equation

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

2006 Vietnam National Olympiad, 1

Tags: logarithms , function , algebra , system of equations , algebra unsolved

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

2008 Moldova MO 11-12, 8

Tags: integration , trigonometry , logarithms , real analysis , real analysis unsolved

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2007 Moldova National Olympiad, 12.3

Tags: inequalities , function , logarithms , algebra proposed , algebra

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

2007 Today's Calculation Of Integral, 184

Tags: calculus , integration , logarithms , function , inequalities , real analysis , calculus computations

(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

2009 Today's Calculation Of Integral, 444

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

Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

1998 Greece JBMO TST, 4

Tags: algebra , polynomial , logarithms , algebra unsolved

(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$. (b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.

1980 AMC 12/AHSME, 18

Tags: trigonometry , logarithms

If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals $\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$

2011 Postal Coaching, 5

Tags: inequalities , logarithms , inequalities unsolved

Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$. Prove that \[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\] for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.

2000 Croatia National Olympiad, Problem 4

Tags: logarithms , algebra , floor function

If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

2002 District Olympiad, 3

Tags: logarithms , calculus , integration , real analysis

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

2013 Today's Calculation Of Integral, 860

Tags: calculus , integration , function , logarithms , analytic geometry , geometry , derivative

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

2012 India Regional Mathematical Olympiad, 3

Tags: logarithms , function , inequalities , inequalities unsolved

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

2007 Harvard-MIT Mathematics Tournament, 33

Tags: calculus , integration , logarithms

Compute \[\int_1^2\dfrac{9x+4}{x^5+3x^2+x}dx.\] (No, your TI-89 doesn’t know how to do this one. Yes, the end is near.)

2014 NIMO Problems, 6

Tags: ceiling function , logarithms , floor function , probability , expected value , number theory , relatively prime

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]