Olympiad problems

2002 Baltic Way, 3

Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that \[a_{m^2+n^2}=a_m^2+a_n^2 \] for all integers $m,n\ge 0$.

PEN J Problems, 11

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

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

2014 Cezar Ivănescu, 1

Tags: inequalities , algebra , logarithm

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $

2012 Today's Calculation Of Integral, 848

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

Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

1959 AMC 12/AHSME, 23

Tags: logarithm , algebra

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

1985 AMC 12/AHSME, 15

Tags: logarithm

If $ a$ and $ b$ are positive numbers such that $ a^b \equal{} b^a$ and $ b \equal{} 9a$, then the value of $ a$ is: $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \sqrt [9] {9}\qquad \textbf{(D)}\ \sqrt [3] {9}\qquad \textbf{(E)}\ \sqrt [4] {3}$

2024 AMC 12/AHSME, 15

Tags: geometry , coordinate geometry , logarithm

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

2003 AMC 12-AHSME, 24

Tags: trigonometry , logarithm

If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$? $ \textbf{(A)}\ \minus{}2 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

1949-56 Chisinau City MO, 40

Tags: algebra , system of equations , logarithm

Solve the system of equations: $$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\ \log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$

2011 Today's Calculation Of Integral, 688

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

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

2024 AIME, 2

Tags: logarithm

Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?

2005 Romania National Olympiad, 3

Tags: function , algebra , logarithm

a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that \[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \] b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$. [i]Mihai Baluna[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

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

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

2005 Today's Calculation Of Integral, 26

Tags: calculus , integration , calculus computations , logarithm

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

1997 AMC 12/AHSME, 17

Tags: logarithm

A line $ x \equal{} k$ intersects the graph of $ y \equal{} \log_5{x}$ and the graph of $ y \equal{} \log_5{(x \plus{} 4)}$. The distance between the points of intersection is $ 0.5$. Given that $ k \equal{} a \plus{} \sqrt{b}$, where $ a$ and $ b$ are integers, what is $ a \plus{} b$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

2010 Contests, 522

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

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

1953 AMC 12/AHSME, 39

Tags: logarithm

The product, $ \log_a b \cdot \log_b a$ is equal to: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ a \qquad\textbf{(C)}\ b \qquad\textbf{(D)}\ ab \qquad\textbf{(E)}\ \text{none of these}$

2011 Today's Calculation Of Integral, 716

Tags: calculus , integration , calculus computations , logarithm

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2004 China Team Selection Test, 2

Tags: floor function , number theory , logarithm

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

2014 Harvard-MIT Mathematics Tournament, 17

Tags: function , induction , logarithm

Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$. (b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?

2018 AMC 12/AHSME, 14

Tags: number theory , logarithm

The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? $\textbf{(A) } 5 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 31 \qquad \textbf{(E) } 35 $

2006 China Second Round Olympiad, 5

Tags: logarithm

Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$

2010 Math Prize For Girls Problems, 17

Tags: function , logarithm

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 $z$?

1998 Vietnam National Olympiad, 1

Tags: integration , limit , logarithm , real analysis

Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.

2014 JBMO Shortlist, 3

Tags: floor function , modular arithmetic , limit , combinatorics , logarithm

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?