Olympiad problems

2012 Today's Calculation Of Integral, 817

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

Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane. Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.

2007 Ukraine Team Selection Test, 3

Tags: algebra , polynomial , logarithms , algebra proposed

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

2011 Regional Competition For Advanced Students, 2

Tags: logarithms , algebra , system of equations , algebra proposed

Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\ \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}

2024 AIME, 2

Tags: AMC , AIME , AIME I , logarithms

Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?

1999 National High School Mathematics League, 3

Tags: logarithms

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

2012 Putnam, 5

Tags: Putnam , function , logarithms , calculus , college contests

Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

1995 AIME Problems, 2

Tags: logarithms , quadratics , modular arithmetic , algebra , polynomial , Vieta , search

Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

2000 Finnish National High School Mathematics Competition, 3

Tags: inequalities , logarithms , induction , algebra unsolved , algebra

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

2013 Purple Comet Problems, 13

Tags: logarithms

There are relatively prime positive integers $m$ and $n$ so that \[\frac{m}{n}=\log_4\left(32^{\log_927}\right).\] Find $m+n$.

2005 Today's Calculation Of Integral, 48

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

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

2000 Tuymaada Olympiad, 4

Tags: inequalities , logarithms , inequalities proposed

Prove for real $x_1$, $x_2$, ....., $x_n$, $0 < x_k \leq {1\over 2}$, the inequality \[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]

2009 Today's Calculation Of Integral, 495

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

Evaluate the following definite integrals. (1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$ (2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$ (3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$

2006 Mathematics for Its Sake, 2

Tags: inequalities , logarithms , algebra

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

1950 AMC 12/AHSME, 37

Tags: logarithms

If $ y \equal{} \log_{a}{x}$, $ a > 1$, which of the following statements is incorrect? $\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ \textbf{(B)}\ \text{If }x=a,y=1 \qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)} \qquad\\ \textbf{(D)}\ \text{If }0<x<z,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero} \qquad\\ \textbf{(E)}\ \text{Only some of the above statements are correct}$

2009 Indonesia TST, 1

Tags: logarithms , induction , inequalities , combinatorics proposed , combinatorics

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

2013 Moldova Team Selection Test, 4

Tags: limit , logarithms , algebra unsolved , algebra

Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

2009 Indonesia TST, 2

Tags: parameterization , logarithms , inequalities , algebra proposed , algebra

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

2012 Today's Calculation Of Integral, 829

Tags: calculus , integration , logarithms , calculus computations

Let $a$ be a positive constant. Find the value of $\ln a$ such that \[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]

1997 IMC, 1

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

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

2004 AMC 12/AHSME, 16

Tags: logarithms , algebra , function , domain

The set of all real numbers $ x$ for which \[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \]is defined is $ \{x|x > c\}$. What is the value of $ c$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$

2007 Today's Calculation Of Integral, 222

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

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

2010 Today's Calculation Of Integral, 568

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

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

1968 AMC 12/AHSME, 23

Tags: logarithms , AMC

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

2012 Pre - Vietnam Mathematical Olympiad, 1

Tags: inequalities , logarithms , calculus , derivative , inequalities proposed

For $a,b,c>0: \; abc=1$ prove that \[a^3+b^3+c^3+6 \ge (a+b+c)^2\]