Olympiad problems

2004 AMC 12/AHSME, 17

Tags: logarithms , Vieta

For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

2007 Today's Calculation Of Integral, 181

Tags: calculus , integration , trigonometry , function , derivative , logarithms , analytic geometry

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

2005 France Team Selection Test, 4

Tags: floor function , ceiling function , induction , inequalities , logarithms , number theory unsolved , number theory

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2006 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , trigonometry , logarithms

Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

2008 Mediterranean Mathematics Olympiad, 4

Tags: algebra , polynomial , induction , logarithms , algebra unsolved

The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$. (a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$. (b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.

1995 Baltic Way, 9

Tags: induction , logarithms , algebra proposed , algebra

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

2010 IMC, 2

Tags: integration , logarithms , real analysis , infinite series

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

1964 AMC 12/AHSME, 11

Tags: logarithms , AMC

Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$. ${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $

2010 Today's Calculation Of Integral, 554

Tags: calculus , integration , logarithms , calculus computations

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

1953 Putnam, A4

Tags: Putnam , logarithms , trigonometry

From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

1976 Miklós Schweitzer, 3

Tags: inequalities , function , logarithms , number theory proposed , number theory

Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that a) $ n! \in H$ for all sufficiently large $ n$, b)$ H$ has density $ 0$. [i]P. Erdos[/i]

2009 Today's Calculation Of Integral, 448

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$

2010 Korea Junior Math Olympiad, 1

Tags: modular arithmetic , logarithms , number theory proposed , number theory

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

2006 All-Russian Olympiad, 5

Tags: logarithms , algebra unsolved , algebra

Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.

2005 Romania National Olympiad, 3

Tags: logarithms , function , algebra proposed , algebra

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]

2010 Contests, 3

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