Olympiad problems

2009 All-Russian Olympiad, 5

Prove that \[ \log_ab\plus{}\log_bc\plus{}\log_ca\le \log_ba\plus{}\log_cb\plus{}\log_ac\] for all $ 1<a\le b\le c$.

2005 Today's Calculation Of Integral, 11

Tags: calculus , integration , logarithms , algebra , function , domain , complex numbers

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

2010 Today's Calculation Of Integral, 630

Tags: calculus , integration , logarithms , calculus computations

Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$

1967 AMC 12/AHSME, 23

Tags: logarithms , limit , AMC

If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{no finite number}$

2007 Putnam, 6

Tags: Putnam , logarithms , integration , probability , analytic geometry , college contests

For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$ \[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]

2005 China National Olympiad, 4

Tags: inequalities , calculus , function , derivative , logarithms , algebra unsolved , algebra

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

2007 AMC 12/AHSME, 17

Tags: logarithms

If $ a$ is a nonzero integer and $ b$ is a positive number such that $ ab^{2} \equal{} \log_{10}b,$ what is the median of the set $ \{0,1,a,b,1/b\}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ a \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \frac {1}{b}$

2011 District Round (Round II), 1

Tags: logarithms , number theory unsolved , number theory

Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?

1985 Traian Lălescu, 1.4

Tags: algebra , equations , logarithms , Floor

Let $ a $ be a non-negative real number distinct from $ 1. $ [b]a)[/b] For which positive values $ x $ the equation $$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$ is true? [b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $

2019 Ramnicean Hope, 3

Tags: floor function , logarithms , algebra

Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor .$ [i]Petre Rău[/i]

2010 AMC 12/AHSME, 12

Tags: logarithms , AMC

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$

2003 AMC 10, 9

Tags: geometry , 3D geometry , logarithms

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$

2010 Math Prize For Girls Problems, 17

Tags: function , logarithms

For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that \[ W(x) e^{W(x)} = x. \] The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that \[ \frac{y}{\log_{2} y} = - \frac{3}{5} \, . \] The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?

Today's calculation of integrals, 898

Tags: calculus , integration , logarithms , calculus computations

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

1986 AMC 12/AHSME, 25

Tags: floor function , logarithms , AMC

If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor = \] $ \textbf{(A)}\ 8192\qquad\textbf{(B)}\ 8204\qquad\textbf{(C)}\ 9218\qquad\textbf{(D)}\ \lfloor \log_{2}(1024!)\rfloor\qquad\textbf{(E)}\ \text{none of these} $

2019 Malaysia National Olympiad, 1

Tags: logarithms , algebra

Evaluate the following sum $$\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}$$

2009 Today's Calculation Of Integral, 450

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

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)}.$

Today's calculation of integrals, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2010 Today's Calculation Of Integral, 560

Tags: calculus , integration , function , geometry , geometric transformation , rotation , logarithms

Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$. (1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis. (2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

2005 Putnam, A5

Tags: Putnam , integration , calculus , trigonometry , logarithms , derivative , function

Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$

2012 Stanford Mathematics Tournament, 3

Tags: logarithms

Given that $\log_{10}2 \approx 0.30103$, ﬁnd the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.

2014 NIMO Problems, 4

Tags: modular arithmetic , Euler , logarithms , floor function , troll

Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

2009 Today's Calculation Of Integral, 456

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

Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$

2014 Contests, 1

Tags: ceiling function , logarithms , combinatorics unsolved , combinatorics

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

2010 Today's Calculation Of Integral, 564

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

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.