Olympiad problems

2023 CCA Math Bonanza, L3.3

Tags: logarithms

Given that $\log_{10}(4) = 0.6021$ to the nearest ten-thousandth, find $\log_{10}(5)$ to the nearest thousandth. [i]Lightning 3.3[/i]

1999 APMO, 2

Tags: logarithms , inequalities , inequalities unsolved , induction

Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i,j=1,2,\dots$. Prove that \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n \] for each positive integer $n$.

2021 JHMT HS, 8

Tags: logarithms , calculus , integration , 2021

Find the unique integer $a > 1$ that satisfies \[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]

2005 China Team Selection Test, 2

Tags: logarithms , algebra unsolved , algebra

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2008 Putnam, B2

Tags: Putnam , logarithms , integration , limit , calculus , factorial , Support

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

2013 Today's Calculation Of Integral, 894

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

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

2009 Today's Calculation Of Integral, 496

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

Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

1967 Miklós Schweitzer, 5

Tags: function , limit , integration , inequalities , geometry , 3D geometry , logarithms

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 \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

2007 IberoAmerican Olympiad For University Students, 3

Tags: calculus , integration , inequalities , function , logarithms , real analysis , Integral inequality

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

1971 IMO Longlists, 44

Tags: induction , logarithms , number theory proposed , number theory

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 $\frac{n}{\nu(n)}=m-1$.

2006 MOP Homework, 1

Tags: ceiling function , logarithms , combinatorics unsolved , combinatorics

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

2007 ITest, 38

Tags: geometry , 3D geometry , floor function , logarithms , inequalities

Find the largest positive integer that is equal to the cube of the sum of its digits.

1954 AMC 12/AHSME, 38

Tags: logarithms

If $ \log 2\equal{}.3010$ and $ \log 3\equal{}.4771$, the value of $ x$ when $ 3^{x\plus{}3}\equal{}135$ is approximately: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 1.47 \qquad \textbf{(C)}\ 1.67 \qquad \textbf{(D)}\ 1.78 \qquad \textbf{(E)}\ 1.63$

1997 Finnish National High School Mathematics Competition, 1

Tags: parameterization , quadratics , logarithms , algebra , polynomial , product of roots , algebra unsolved

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2007 Romania National Olympiad, 1

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

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

1998 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , limit , logarithms , trigonometry

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

2009 Princeton University Math Competition, 2

Tags: calculus , integration , floor function , logarithms , induction , ceiling function

It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2020 AMC 12/AHSME, 13

Tags: AMC 12 B , AMC , AMC 12 , logarithms , 2020 AMC 12B , 2020 AMC

Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$ $\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$

2011 International Zhautykov Olympiad, 3

Tags: modular arithmetic , logarithms , induction , number theory unsolved , number theory

Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.

2017 AIME Problems, 7

Tags: logarithms , AMC , AIME , AIME II , genius title

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.

2022 Romania National Olympiad, P1

Tags: logarithms , algebra , romania

Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ [i]Mihai Opincariu[/i]

2010 Today's Calculation Of Integral, 539

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

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx$.

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: algebra , logarithms

Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

2012 Purple Comet Problems, 9

Tags: logarithms

Find the value of $x$ that satisfies $\log_{3}(\log_9x)=\log_9(\log_3x)$

2008 AMC 12/AHSME, 16

Tags: logarithms , arithmetic sequence , AMC

The numbers $ \log(a^3b^7)$, $ \log(a^5b^{12})$, and $ \log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $ 12^\text{th}$ term of the sequence is $ \log{b^n}$. What is $ n$? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 112 \qquad \textbf{(E)}\ 143$