Found problems: 1187
A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D\rfloor$? (For real $x$, $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)
Find all pairs $(x,y)$ of positive real numbers such that $xy$ is an integer and $x+y = \lfloor x^2 - y^2 \rfloor$.
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ \text{Infinitely many}
\qquad\textbf{e)}\ \text{None of above}
$
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]
Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$
If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following:
$$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation
$$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$
Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation
$$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$
where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1).
Note: $[x]$ indicates the whole part of the number $x$.
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.
Solve in the set $R$ the equation
$$2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}$$
where $[x]$ and $\{x\}$ represent the integer part and the fractional part of the real number $x$, respectively.
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.
Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ?
$[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.
Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor$.
Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$.
(Peter Cameron)
Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.
Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]
For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[
X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...).
\] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
Find all pairs $(x, y)$ of real numbers such that
\[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\]
where $f(x)=[x]$ is the floor function.
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by David Altizio[/i]
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.