Found problems: 1340
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$
Find all real numbers $x$, such that:
a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$
b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$
Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
Let $ \alpha$ be the positive root of the equation $ x^2 \minus{} 1989x \minus{} 1 \equal{} 0.$ Prove that there exist infinitely many natural numbers $ n$ that satisfy the equation:
\[ \lfloor \alpha n \plus{} 1989 \alpha \lfloor \alpha n \rfloor \rfloor \equal{} 1989n \plus{} \left( 1989^2 \plus{} 1 \right) \lfloor \alpha n \rfloor.\]
Find the greatest integer less than the number
$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits.
[img]http://i2.tinypic.com/5x73dza.png[/img]
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $
[i]Constantin Nicolau[/i]
Solve the equation:
$[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number.
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).
Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]
Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?
Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.
A rational number $ x$ is called [i]good[/i] if it satisfies: $ x\equal{}\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)\equal{}1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)}\] Find all the good numbers.
Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ).
Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
The $ 2007$ AMC $ 10$ will be scored by awarding $ 6$ points for each correct response, $ 0$ points for each incorrect response, and $ 1.5$ points for each problem left unanswered. After looking over the $ 25$ problems, Sarah has decided to attempt the first $ 22$ and leave only the last $ 3$ unanswered. How many of the first $ 22$ problems must she solve correctly in order to score at least $ 100$ points?
$ \textbf{(A)}\ 13\qquad
\textbf{(B)}\ 14\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 16\qquad
\textbf{(E)}\ 17$
Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied :
\[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\]
for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.
As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all.
Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?"
"Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines.
Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.
Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.