Olympiad problems

2009 China Team Selection Test, 1

Tags: logarithm , ceiling function , function , combinatorics , ratio , floor function

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

2009 Today's Calculation Of Integral, 450

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

Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$

2004 AMC 12/AHSME, 17

Tags: logarithm , vieta

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$

1999 Brazil Team Selection Test, Problem 4

Tags: logarithm , triangle inequality , ring theory , function , number theory , induction , inequalities

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

1995 Irish Math Olympiad, 1

Tags: logarithm , inequalities , induction

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

2007 Putnam, 3

Tags: induction , logarithm , function , calculus , integration , floor function

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

2015 AMC 12/AHSME, 8

Tags: logarithm

What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$? $\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$

1982 Tournament Of Towns, (027) 1

Tags: radical , logarithm , sum , algebra , floor function

Prove that for all natural numbers $n$ greater than $1$ : $$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$ (VV Kisil)

2010 Malaysia National Olympiad, 2

Tags: logarithm , algebra

Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]

2008 Putnam, A4

Tags: integration , logarithm , floor function

Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?

2010 AMC 12/AHSME, 12

Tags: logarithm

For what value of $ x$ does \[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\] $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$

2014 NIMO Problems, 1

Tags: modular arithmetic , geometry , logarithm , 3d geometry , prime factorization , number theory , function

Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$. [i]Proposed by Kevin Sun[/i]

2000 AIME Problems, 1

Tags: logarithm , relatively prime , number theory

The number \[ \frac 2{\log_4{2000^6}}+\frac 3{\log_5{2000^6}} \] can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2011 Today's Calculation Of Integral, 717

Tags: geometry , logarithm , calculus computations , integration , calculus

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

2020 AMC 12/AHSME, 10

Tags: logarithm

There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\] What is the sum of the digits of $n?$ $\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

2009 Unirea, 4

Tags: geometry , limit , logarithm , integration , trigonometry , probability , modular arithmetic

Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

2006 IMC, 3

Tags: logarithm , trigonometry

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

2009 Today's Calculation Of Integral, 447

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

Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

1970 Czech and Slovak Olympiad III A, 6

Tags: inequalities , logarithm , algebra , trigonometry

Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]

1993 Brazil National Olympiad, 5

Tags: logarithm , algebra , functional equation , function

Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.

2006 AIME Problems, 9

Tags: logarithm , ratio , modular arithmetic

The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$.

2024 AMC 12/AHSME, 15

Tags: geometry , logarithm , coordinate geometry

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

2009 Today's Calculation Of Integral, 448

Tags: integration , logarithm , calculus , calculus computations

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

1962 AMC 12/AHSME, 28

Tags: logarithm

The set of $ x$-values satisfying the equation $ x^{\log_{10} x} \equal{} \frac{x^3}{100}$ consists of: $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \text{10, only} \qquad \textbf{(C)}\ \text{100, only} \qquad \textbf{(D)}\ \text{10 or 100, only} \qquad \textbf{(E)}\ \text{more than two real numbers.}$

1991 Arnold's Trivium, 5

Tags: calculus , derivative , logarithm , function

Calculate the $100$th derivative of the function \[\frac{1}{x^2+3x+2}\] at $x=0$ with $10\%$ relative error.