Olympiad problems

2006 IMC, 3

Tags: trigonometry , logarithm

Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in \left]0,\frac{\pi}{2}\right[$.

2005 Iran MO (3rd Round), 3

Tags: geometry , rectangle , combinatorics , logarithm

$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino. Prove that $10000\leq f(1384)\leq960000$. Find some bound for $f(n)$

2005 Today's Calculation Of Integral, 1

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

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] $\int \cos ^4 x dx$

2005 Today's Calculation Of Integral, 74

Tags: calculus , integration , calculus computations , logarithm

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

2007 Tuymaada Olympiad, 4

Tags: probability , limit , ratio , greatest common divisor , number theory , prime number , logarithm

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

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 Today's Calculation Of Integral, 830

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

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

2009 Iran Team Selection Test, 12

Tags: combinatorics , logarithm

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

2025 District Olympiad, P1

Tags: logarithm

Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$ [i]Mathematical Gazette[/i]

2009 Today's Calculation Of Integral, 446

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2015 AMC 12/AHSME, 14

Tags: logarithm

What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$

1953 AMC 12/AHSME, 22

Tags: logarithm

The logarithm of $ 27\sqrt[4]{9}\sqrt[3]{9}$ to the base $ 3$ is: $ \textbf{(A)}\ 8\frac{1}{2} \qquad\textbf{(B)}\ 4\frac{1}{6} \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these}$

2004 Romania Team Selection Test, 4

Tags: geometry , circumcircle , limit , gauss , algebra , polynomial , logarithm

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

1969 AMC 12/AHSME, 17

Tags: logarithm

The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by: $\textbf{(A) }\log3\qquad \textbf{(B) }\tfrac12\log6\qquad \textbf{(C) }1+\log\tfrac34\qquad$ $\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad \textbf{(E) }\text{none of these}$

1973 Canada National Olympiad, 1

Tags: inequalities , calculus , integration , logarithm

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

1950 AMC 12/AHSME, 25

Tags: logarithm

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

2010 Today's Calculation Of Integral, 660

Tags: calculus , integration , calculus computations , logarithm

Let $a,\ b$ be given positive constants. Evaluate \[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\] Own

1977 AMC 12/AHSME, 18

Tags: logarithm

If $y=(\log_23)(\log_34)\cdots(\log_n[n+1])\cdots(\log_{31}32)$ then $\textbf{(A) }4<y<5\qquad\textbf{(B) }y=5\qquad\textbf{(C) }5<y<6\qquad$ $\textbf{(D) }y=6\qquad \textbf{(E) }6<y<7$

2007 Today's Calculation Of Integral, 205

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

2004 AMC 12/AHSME, 17

Tags: vieta , logarithm

For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

2012 Stanford Mathematics Tournament, 10

Tags: probability , ceiling function , logarithm

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

2010 Today's Calculation Of Integral, 540

Tags: calculus , integration , calculus computations , logarithm

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

1989 IMO Longlists, 45

Tags: number theory , algebra , logarithm

Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.

1973 Miklós Schweitzer, 9

Tags: probability and stats , logarithm

Determine the value of \[ \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi],\] where $ \xi$ is a random variable and $ E$ denotes expectation. [i]Z. Daroczy[/i]

PEN A Problems, 25

Tags: number theory , least common multiple , floor function , modular arithmetic , function , inequalities , logarithm

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.