Olympiad problems

1953 AMC 12/AHSME, 22

Tags: logarithms

The logarithm of $ 27\sqrt[4]{9}\sqrt[3]{9}$ to the base $ 3$ is: $ \textbf{(A)}\ 8\frac{1}{2} \qquad\textbf{(B)}\ 4\frac{1}{6} \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these}$

2009 Today's Calculation Of Integral, 445

Tags: calculus , integration , logarithms , 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.$

2014 Singapore Senior Math Olympiad, 3

Tags: logarithms

Find the value of $\frac{\log_59\log_75\log_37}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}}$ $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7 $

1977 Miklós Schweitzer, 6

Tags: function , logarithms , real analysis , real analysis unsolved

Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$. [i]Z. Daroczy, Gy. Maksa[/i]

1973 Miklós Schweitzer, 9

Tags: logarithms , probability and stats

Determine the value of \[ \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi],\] where $ \xi$ is a random variable and $ E$ denotes expectation. [i]Z. Daroczy[/i]

2011 ELMO Shortlist, 5

Tags: logarithms , algorithm , combinatorics proposed , combinatorics

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

2004 China Team Selection Test, 2

Tags: floor function , logarithms , number theory

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 Singapore Senior Math Olympiad, 11

Tags: logarithms

Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$

2007 Baltic Way, 1

Tags: inequalities , quadratics , function , logarithms , induction , rearrangement inequality , algebra proposed

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]

1964 AMC 12/AHSME, 21

Tags: logarithms , quadratics , AMC

If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals: $ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $

1996 AIME Problems, 2

Tags: floor function , logarithms

For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.

2010 Today's Calculation Of Integral, 616

Tags: calculus , integration , logarithms , calculus computations

Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

2013 AIME Problems, 2

Tags: logarithms , AMC , algebra , AIME

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

2014 AIME Problems, 7

Tags: trigonometry , logarithms , AMC , sums , AIME , AIME II , 2014 AIME II

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

2009 AIME Problems, 7

Tags: logarithms , induction , invariant , AMC

The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , logarithms , inequalities , calculus

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

1972 AMC 12/AHSME, 6

Tags: logarithms , AMC

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

2007 Princeton University Math Competition, 5

Tags: logarithms

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

2010 Today's Calculation Of Integral, 581

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

For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.

1994 Turkey Team Selection Test, 2

Tags: limit , logarithms , algebra proposed , algebra

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

2010 AIME Problems, 14

Tags: floor function , logarithms , search , inequalities , function , algebra , AMC

For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$. [b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.

2009 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , function , integration , logarithms

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

2012 Balkan MO Shortlist, C1

Tags: induction , inequalities , floor function , logarithms , combinatorics unsolved , combinatorics

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2010 Today's Calculation Of Integral, 573

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

Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

2012 Today's Calculation Of Integral, 837

Tags: calculus , integration , limit , logarithms , calculus computations

Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$ Find $\lim_{n\to\infty} f_n(1).$