Found problems: 1187
For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms?
$\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105$
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $
Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $
[i]Dumitru Acu[/i]
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.\]
For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.
For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.
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.
Given a square-free integer $n>2$, evaluate the sum $\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor$.
Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.
Determine all integers $x$ satisfying
\[ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. \]
($[y]$ is the largest integer which is not larger than $y.$)
Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.
А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$.
[i]Alexey Tolpygo[/i]
Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ :
\[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
Let $S$ be a $k$-element set.
[i](a)[/i] Find the number of mappings $f : S \to S$ such that
\[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\]
[i](b)[/i] The same with the condition $\text{(i)}$ left out.
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
Prove that $ \lfloor{(4+\sqrt11)^{n}}\rfloor $ is odd for every natural number n.
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $[M]$.
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]
[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]