Olympiad problems

1969 Miklós Schweitzer, 4

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]

2013 AIME Problems, 2

Tags: algebra , logarithm

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

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$.

2012 Romania Team Selection Test, 1

Tags: floor function , function , induction , algebra , logarithm

Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]

2007 Purple Comet Problems, 12

Tags: logarithm

Find the maximum possible value of $8\cdot 27^{\log_6 x}+27\cdot 8^{\log_6 x}-x^3$ as $x$ varies over the positive real numbers.

2011 ELMO Shortlist, 5

Tags: algorithm , combinatorics , logarithm

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

1998 AMC 12/AHSME, 22

Tags: geometric series , logarithm

What is the value of the expression \[ \frac {1}{\log_2 100!} \plus{} \frac {1}{\log_3 100!} \plus{} \frac {1}{\log_4 100!} \plus{} \cdots \plus{} \frac {1}{\log_{100} 100!}? \]$ \textbf{(A)}\ 0.01 \qquad \textbf{(B)}\ 0.1 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 10$

1973 Miklós Schweitzer, 3

Tags: floor function , prime number , number theory , logarithm

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

2013 Harvard-MIT Mathematics Tournament, 35

Tags: hmmt , floor function , function , geometry , geometric transformation , reflection , logarithm

Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.

2018 District Olympiad, 3

Tags: inequalities , logarithm

Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and \[\log_ab + 2\log_bc + 5\log_ca = 12.\] Prove that \[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

1979 IMO Shortlist, 14

Tags: logarithm

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

2005 Today's Calculation Of Integral, 2

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

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

1951 AMC 12/AHSME, 45

Tags: logarithm

If you are given $ \log 8 \approx .9031$ and $ \log 9 \approx .9542$, then the only logarithm that cannot be found without the use of tables is: $ \textbf{(A)}\ \log 17 \qquad\textbf{(B)}\ \log \frac {5}{4} \qquad\textbf{(C)}\ \log 15 \qquad\textbf{(D)}\ \log 600 \qquad\textbf{(E)}\ \log .4$

2005 Today's Calculation Of Integral, 18

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

Calculate the following indefinite integrals. [1] $\int (\sin x+\cos x)^4 dx$ [2] $\int \frac{e^{2x}}{e^x+1}dx$ [3] $\int \sin ^ 4 xdx$ [4] $\int \sin 6x\cos 2xdx$ [5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$

2014 China National Olympiad, 1

Tags: induction , algebra , number theory , logarithm

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.

2013 ELMO Shortlist, 2

Tags: induction , strong induction , combinatorics , logarithm

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

2012 Today's Calculation Of Integral, 785

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

For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$

1957 AMC 12/AHSME, 5

Tags: logarithm

Through the use of theorems on logarithms \[ \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}}\] can be reduced to: $ \textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ 0\qquad \textbf{(E)}\ \log{\frac{a^2y}{d^2x}}$

2002 APMO, 1

Tags: floor function , induction , function , inequalities , calculus , derivative , logarithm

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

2012 AIME Problems, 9

Tags: number theory , relatively prime , logarithm

Let $x$, $y$, and $z$ be positive real numbers that satisfy \[ 2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0. \] The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{p/q}}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.

2011 China Second Round Olympiad, 9

Tags: logarithm , algebra

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

2017 AIME Problems, 7

Tags: genius title , logarithm

Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.

2005 AMC 12/AHSME, 23

Tags: probability , floor function , logarithm

Two distinct numbers $ a$ and $ b$ are chosen randomly from the set $ \{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $ \log_{a} b$ is an integer? $ \textbf{(A)}\ \frac {2}{25} \qquad \textbf{(B)}\ \frac {31}{300} \qquad \textbf{(C)}\ \frac {13}{100} \qquad \textbf{(D)}\ \frac {7}{50} \qquad \textbf{(E)}\ \frac {1}{2}$

2010 Today's Calculation Of Integral, 607

Tags: calculus , integration , analytic geometry , geometry , function , parameterization , logarithm

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.