Olympiad problems

1994 Vietnam National Olympiad, 1

Find all real solutions to \[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\] \[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\] \[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]

1980 Swedish Mathematical Competition, 1

Tags: irrational , logarithm , algebra

Show that $\log_{10} 2$ is irrational.

1973 Miklós Schweitzer, 5

Tags: limit , integration , logarithm , real analysis

Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\] [i]P. Medgyessy[/i]

2004 Putnam, B2

Tags: inequalities , probability , function , combinatorics , logarithm

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

1991 Arnold's Trivium, 66

Tags: complex analysis , complex number , logarithm

Solve the Dirichlet problem \[\Delta u=0\text{ for }x^2+y^2<1\] \[u=1\text{ for }x^2+y^2=1,\;y>0\] \[u=-1\text{ for }x^2+y^2=1,\;y<0\]

1997 IMC, 1

Tags: limit , calculus , integration , logarithm , real analysis

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

2017 District Olympiad, 2

Tags: equation , system , logarithm

Solve in $ \mathbb{Z} $ the system: $$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$

2011 Today's Calculation Of Integral, 710

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

Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2011 Regional Competition For Advanced Students, 2

Tags: system of equations , algebra , logarithm

Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\ \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}

2013 Today's Calculation Of Integral, 871

Tags: calculus , integration , trigonometry , limit , calculus computations , definite integral , logarithm

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

1997 APMO, 5

Tags: algorithm , rotation , geometry , geometric transformation , combinatorics , logarithm

Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that \[ a_1 + a_2 + \cdots + a_n = nN \] where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.) How should this redistribution be performed so that the total number of objects transferred is minimum?

1994 APMO, 5

Tags: floor function , number theory , logarithm

You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

PEN O Problems, 40

Tags: floor function , induction , logarithm

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

2022 Romania National Olympiad, P1

Tags: algebra , logarithm

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]

2007 Today's Calculation Of Integral, 179

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2008 Putnam, A4

Tags: integration , floor function , logarithm

Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?

2005 Today's Calculation Of Integral, 46

Tags: calculus , integration , calculus computations , logarithm

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

1955 AMC 12/AHSME, 17

Tags: logarithm

If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals: $ \textbf{(A)}\ 1.25 \qquad \textbf{(B)}\ 0.81 \qquad \textbf{(C)}\ 2.43 \qquad \textbf{(D)}\ 0.8 \qquad \textbf{(E)}\ \text{either 0.8 or 1.25}$

1987 India National Olympiad, 1

Tags: number theory , relatively prime , algebra , logarithm

Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that \[ \frac{\log_{10} m}{\log_{10} n}\] is not a rational number.

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.

1998 USAMTS Problems, 1

Tags: logarithm

Determine the leftmost three digits of the number \[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , logarithm

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

2005 Today's Calculation Of Integral, 26

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

1987 IMO Longlists, 17

Tags: algebra , logarithm

Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.