Olympiad problems

2003 AMC 10, 9

Find the value of $ x$ that satisfies the equation \[ 25^{\minus{}2}\equal{}\frac{5^{48/x}}{5^{26/x}\cdot25^{17/x}}. \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

1996 Brazil National Olympiad, 3

Tags: combinatorics , logarithm

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

2010 Today's Calculation Of Integral, 584

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

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

1960 AMC 12/AHSME, 24

Tags: geometry , 3d geometry , calculus , integration , function , irrational number , logarithm

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

2004 India IMO Training Camp, 1

Tags: function , logarithm , inequalities

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

2008 ISI B.Stat Entrance Exam, 6

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

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2007 Putnam, 4

Tags: algebra , polynomial , quadratic , logarithm

A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$

1966 AMC 12/AHSME, 16

Tags: logarithm

If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ are real numbers, then $xy$ equals: $\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$

1982 Miklós Schweitzer, 4

Tags: number theory , logarithm

Let \[ f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .\] Prove that \[ \limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .\] [i]P. Erdos[/i]

2021 AMC 12/AHSME Fall, 9

Tags: logarithm

A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_2 x$, $\log_3 x$, and $\log_4 x$. What is $x$? $\textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }6\sqrt{6}\qquad\textbf{(C) }24\qquad\textbf{(D) }48\qquad\textbf{(E) }576$

2021 JHMT HS, 9

Tags: algebra , logarithm

Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\tfrac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

2009 Princeton University Math Competition, 3

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

It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2008 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration , calculus computations , logarithm

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2009 AMC 12/AHSME, 24

Tags: function , inequalities , logarithm

The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that \[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}} \]is defined? $ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

1995 Baltic Way, 9

Tags: induction , logarithm , algebra

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

1992 Hungary-Israel Binational, 3

Tags: number theory , logarithm

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there inﬁnitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$

2012 China Team Selection Test, 1

Tags: floor function , function , number theory , logarithm

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

Today's calculation of integrals, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2005 Today's Calculation Of Integral, 37

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

Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

1977 Miklós Schweitzer, 3

Tags: superior algebra , logarithm

Prove that if $ a,x,y$ are $ p$-adic integers different from $ 0$ and $ p | x, pa | xy$, then \[ \frac 1y \frac{(1\plus{}x)^y\minus{}1}{x} \equiv \frac{\log (1\plus{}x)}{x} \;\;\;\; ( \textrm{mod} \; a\ ) \\\\ .\] [i]L. Redei[/i]

MathLinks Contest 7th, 5.3

Tags: inequalities , symmetry , function , calculus , derivative , parameterization , logarithm

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

2001 India National Olympiad, 3

Tags: inequalities , ineqstd , logarithm

If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that \[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]

1974 Miklós Schweitzer, 6

Tags: induction , function , integration , limit , logarithm , real analysis

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

2013 ELMO Shortlist, 9

Tags: function , logarithm , inequalities

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]