Found problems: 1187
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)
Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).
An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer.
[i]A. Golovanov[/i]
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
Prove that the following inequality holds for all real $a$ and natural $n$: $$|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$$ $F(a)$ is the distance from $a$ to the closest integer.
There are several scientists collaborating in Niichavo. During an $ 8$-hour working day, the scientists went to cafeteria, possibly several times.It is known that for every two scientist, the total time in which exactly one of them was in cafeteria is at least $ x$ hours ($ x>4$).
What is the largest possible number of scientist that could work in Niichavo that day,in terms of $ x$?
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that
\[0<|xy-zw|<C\alpha ^{-4}\]
where $\alpha =\frac{|X|}{n}$.
Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.
Two distinct numbers $ a$ and $ b$ are chosen randomly from the set $ \{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $ \log_{a} b$ is an integer?
$ \textbf{(A)}\ \frac {2}{25} \qquad \textbf{(B)}\ \frac {31}{300} \qquad \textbf{(C)}\ \frac {13}{100} \qquad \textbf{(D)}\ \frac {7}{50} \qquad \textbf{(E)}\ \frac {1}{2}$
A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$.
a) Prove that $\frac{s}{r}$ is a natural number.
b) Show that both $r$ and $s$ are natural numbers.
Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.
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.
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded?
[b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $
Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $
[i]Mihai Piticari[/i] and [i]Dan Popescu[/i]
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.
Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$
Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).
Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer.
[i]Dan Schwarz, Romania[/i]
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$
Prove that at least $n - 1$ of these numbers are perfect squares.
Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.
For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.
Show that a segment of length $ h$ can go through or be tangent to at most $ 2\lfloor h/\sqrt{2}\rfloor\plus{}2$ nonoverlapping unit
spheres.
[i]L.Fejes-Toth, A. Heppes[/i]
What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?
Let $a, b$ and $c$ be real numbers such that
$$\lceil a \rceil + \lceil b \rceil + \lceil c \rceil + \lfloor a + b \rfloor + \lfloor b + c \rfloor + \lfloor c + a \rfloor = 2020$$
Prove that
$$\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor + \lceil a + b + c \rceil \ge 1346$$
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. That is, $\lfloor x \rfloor$ is the unique integer satisfying $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$, and $\lceil x \rceil$ is the unique integer satisfying $\lceil x \rceil - 1 < x \le \lceil x \rceil$.
[i]Proposed by Ariel García[/i]
Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$.