Olympiad problems

2019 AIME Problems, 6

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down \begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54 \end{align*} and finds that this system of equations has a single real number solution $x > 1$. Find $b$.

1972 AMC 12/AHSME, 6

Tags: logarithm

If $3^{2x}+9=10(3^{x})$, then the value of $(x^2+1)$ is $\textbf{(A) }1\text{ only}\qquad\textbf{(B) }5\text{ only}\qquad\textbf{(C) }1\text{ or }5\qquad\textbf{(D) }2\qquad \textbf{(E) }10$

2005 Today's Calculation Of Integral, 74

Tags: calculus , integration , calculus computations , logarithm

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

1967 Miklós Schweitzer, 5

Tags: function , limit , integration , inequalities , geometry , 3d geometry , logarithm

Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

2011 Today's Calculation Of Integral, 763

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

Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$

1974 Canada National Olympiad, 1

Tags: induction , logarithm

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

1983 AMC 12/AHSME, 12

Tags: logarithm

If $\log_7 \Big(\log_3 (\log_2 x) \Big) = 0$, then $x^{-1/2}$ equals $\displaystyle \text{(A)} \ \frac{1}{3} \qquad \text{(B)} \ \frac{1}{2 \sqrt 3} \qquad \text{(C)} \ \frac{1}{3 \sqrt 3} \qquad \text{(D)} \ \frac{1}{\sqrt{42}} \qquad \text{(E)} \ \text{none of these}$

1997 Vietnam Team Selection Test, 2

Tags: algebra , logarithm

Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.

Today's calculation of integrals, 883

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

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

2004 AIME Problems, 12

Tags: geometry , floor function , inequalities , rectangle , number theory , relatively prime , logarithm

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

1998 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , integration , function , ratio , geometric series , logarithm

Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.

2014 AIME Problems, 7

Tags: trigonometry , logarithm , sum

Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]

2000 AMC 12/AHSME, 7

Tags: geometry , 3d geometry , logarithm

How many positive integers $ b$ have the property that $ \log_b729$ is a positive integer? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

1988 AIME Problems, 3

Tags: logarithm

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

1990 AMC 12/AHSME, 23

Tags: logarithm

If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ $ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $

2023 AMC 12/AHSME, 7

Tags: logarithm

For how many integers $n$ does the expression \[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm? $ \textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

1999 Federal Competition For Advanced Students, Part 2, 1

Tags: floor function , number theory , logarithm

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

2010 Contests, 3

Tags: induction , floor function , logarithm , combinatorics

There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

2013 Today's Calculation Of Integral, 887

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

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2006 Iran Team Selection Test, 2

Tags: ceiling function , logarithm , algebra

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

2002 Putnam, 6

Tags: geometry , integration , function , logarithm

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?

2020 Kosovo National Mathematical Olympiad, 3

Tags: algebra , minimum value , logarithm

Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

2021 JHMT HS, 9

Tags: logarithm , algebra

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 $m + n$.

2004 Nicolae Coculescu, 2

Tags: quadratic , algebra , trigonometry , logarithm , equation , monotone

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

1950 AMC 12/AHSME, 18

Tags: logarithm

Of the following (1) $ a(x\minus{}y)\equal{}ax\minus{}ay$ (2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$ (3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$ (4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$ (5) $ a(xy)\equal{}ax\times ay$ $\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\ \textbf{(E)}\ \text{Only 1 is true}$