Olympiad problems

1971 IMO Longlists, 44

Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.

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

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: $\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$

2020 IMC, 8

Tags: real analysis , logarithm , binomial coefficient

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

2006 ISI B.Stat Entrance Exam, 8

Tags: limit , algebra , logarithm

Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that \[\frac{\log_2x}{x}=c\] has only one real solution.

1951 AMC 12/AHSME, 22

Tags: quadratic , algebra , quadratic formula , logarithm

The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are: $ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$ $ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$

2005 AIME Problems, 8

Tags: vieta , algebra , polynomial , number theory , logarithm

The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2013 Korea - Final Round, 3

Tags: floor function , inequalities , logarithm

For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of \[ \sum_{(i,j) \in T} x_i x_j \] in terms of $n$.

1969 AMC 12/AHSME, 17

Tags: logarithm

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

1968 AMC 12/AHSME, 23

Tags: logarithm

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

1991 Arnold's Trivium, 5

Tags: calculus , derivative , function , logarithm

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

PEN A Problems, 13

Tags: floor function , inequalities , divisibility theory , logarithm

Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

2004 Romania National Olympiad, 2

Tags: function , algebra , polynomial , calculus , integration , inequalities , logarithm

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

2005 Today's Calculation Of Integral, 6

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

Calculate the following indefinite integrals. [1] $\int \sin x\cos ^ 3 x dx$ [2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$ [3] $\int x^2 \sqrt{x^3+1}dx$ [4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$ [5] $\int (1-x^2)e^x dx$

Today's calculation of integrals, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

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 $n$.

1987 India National Olympiad, 2

Tags: algebra , logarithm

Determine the largest number in the infinite sequence \[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]

2014 ELMO Shortlist, 9

Tags: number theory , logarithm

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

2010 Today's Calculation Of Integral, 626

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

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

2013 Waseda University Entrance Examination, 3

Tags: function , calculus , calculus computations , logarithm

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

1979 IMO Longlists, 37

Tags: logarithm

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

2010 Today's Calculation Of Integral, 568

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

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}$

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 $x$? $\textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }6\sqrt{6}\qquad\textbf{(C) }24\qquad\textbf{(D) }48\qquad\textbf{(E) }576$

2009 Putnam, A5

Tags: abstract algebra , group theory , logarithm

Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$