Olympiad problems

2016 AIME Problems, 3

Tags: logarithm

Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x) &= 5 \\ \log_3(xyz-3+\log_5 y) &= 4 \\ \log_4(xyz-3+\log_5 z) &= 4. \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.

1997 National High School Mathematics League, 12

Tags: logarithm

Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.

2010 Today's Calculation Of Integral, 613

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

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

2013 Today's Calculation Of Integral, 898

Tags: integration , logarithm , calculus , calculus computations

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

2013 Miklós Schweitzer, 12

Tags: probability and stats , ratio , logarithm , inequalities

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let \[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\] Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$ [i]Proposed by Tamás Móri[/i]

2010 Today's Calculation Of Integral, 636

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

Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$ (1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$. (2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$. [i]1992 Tokyo University entrance exam/Science[/i]

2009 Today's Calculation Of Integral, 444

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

Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

2005 AMC 12/AHSME, 13

Tags: logarithm

Suppose that $ 4^{x_1} \equal{} 5, 5^{x_2} \equal{} 6, 6^{x_3} \equal{} 7,...,127^{x_{124}} \equal{} 128$. What is $ x_1x_2 \cdots x_{124}$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ \frac {5}{2}\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ \frac {7}{2}\qquad \textbf{(E)}\ 4$

2012 AIME Problems, 9

Tags: logarithm , relatively prime , number theory

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 $p+q$.

1995 Israel Mathematical Olympiad, 1

Tags: algebra , logarithm , system of equations

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

2000 National High School Mathematics League, 9

Tags: logarithm , geometric series , ratio

If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.

2013 Today's Calculation Of Integral, 889

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

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

1966 IMO Shortlist, 30

Tags: logarithm , inequalities , calculus

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2007 Princeton University Math Competition, 10

Tags: logarithm

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

1973 Canada National Olympiad, 1

Tags: integration , logarithm , inequalities , calculus

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , logarithm , inequalities , calculus

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

PEN J Problems, 11

Tags: function , inequalities , logarithm , floor function , modular arithmetic , number theory , induction

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

2006 AIME Problems, 2

Tags: logarithm , triangle inequality , geometry

The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.

2014 Contests, 3

Tags: parallelogram , trapezoid , geometric transformation , logarithm , dilation , trigonometry , geometry

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

1953 AMC 12/AHSME, 47

Tags: inequalities , logarithm , calculus

If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2011 IMC, 3

Tags: limit , logarithm

Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.

2010 AIME Problems, 5

Tags: logarithm , sfft , special factorizations

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

2009 Today's Calculation Of Integral, 445

Tags: logarithm , integration , calculus , calculus computations

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2007 Today's Calculation Of Integral, 205

Tags: logarithm , integration , calculus , calculus computations

Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

2002 Putnam, 6

Tags: logarithm , integration , geometry , function

Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does \[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} \] converge?