Found problems: 1187
PEN M Problems, 2
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.
II Soros Olympiad 1995 - 96 (Russia), 10.4
Solve the system of equations
$$\begin{cases} x^2+ [y]=10
\\ y^2+[x]=13
\end{cases}$$
($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).
1991 AIME Problems, 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
1989 All Soviet Union Mathematical Olympiad, 503
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.
2013 Online Math Open Problems, 23
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$?
[hide="Clarifications"]
[list]
[*] $S$ is the ``set of 10 distinct integers'' from the first sentence.[/list][/hide]
[i]Ray Li[/i]
2018 Korea Winter Program Practice Test, 1
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
1984 AIME Problems, 12
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$?
1976 IMO Longlists, 29
Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula
if
\[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \]
Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$
2024 Stars of Mathematics, P1
Fix a positive integer $n\geq 2$. What is the lest value that the expression $$\bigg\lfloor\frac{x_2+x_3+\dots +x_n}{x_1}\bigg\rfloor + \bigg\lfloor\frac{x_1+x_3+\dots +x_n}{x_2}\bigg\rfloor +\dots +\bigg\lfloor\frac{x_1+x_2+\dots +x_{n-1}}{x_n}\bigg\rfloor$$ may achieve, where $x_1,x_2,\dots ,x_n$ are positive real numbers.
PEN I Problems, 2
Prove that for any positive integer $n$, \[\left\lfloor \frac{n}{3}\right\rfloor+\left\lfloor \frac{n+2}{6}\right\rfloor+\left\lfloor \frac{n+4}{6}\right\rfloor = \left\lfloor \frac{n}{2}\right\rfloor+\left\lfloor \frac{n+3}{6}\right\rfloor .\]
2006 Austrian-Polish Competition, 7
Find all nonnegative integers $m,n$ so that \[\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}\]
2020 Kazakhstan National Olympiad, 1
There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times.
[i]Proposed by T. Zimanov[/i]
2016 India PRMO, 10
Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.
2012 Vietnam Team Selection Test, 3
Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition:
For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.
2013 Romanian Master of Mathematics, 2
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
2010 Belarus Team Selection Test, 4.3
a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$
b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich)
2011 Stars Of Mathematics, 4
Let $n\geq 2$ be an integer. Let us call [i]interval[/i] a subset $A \subseteq \{1,2,\ldots,n\}$ for which integers $1\leq a < b\leq n$ do exist, such that $A = \{a,a+1,\ldots,b-1,b\}$. Let a family $\mathcal{A}$ of subsets $A_i \subseteq \{1,2,\ldots,n\}$, with $1\leq i \leq N$, be such that for any $1\leq i < j \leq N$ we have $A_i \cap A_j$ being an interval.
Prove that $\displaystyle N \leq \left \lfloor n^2/4 \right \rfloor$, and that this bound is sharp.
(Dan Schwarz - after an idea by Ron Graham)
2006 AIME Problems, 13
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?
2007 Regional Competition For Advanced Students, 2
Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and
$ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$
2004 National Olympiad First Round, 10
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$, $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$. What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$? ($\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$)
$
\textbf{(A)}\ 2005
\qquad\textbf{(B)}\ 2006
\qquad\textbf{(C)}\ 2007
\qquad\textbf{(D)}\ 2008
\qquad\textbf{(E)}\ \text{None of above}
$
2002 Tournament Of Towns, 7
Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?
2004 South East Mathematical Olympiad, 7
A tournament is held among $n$ teams, following such rules:
a) every team plays all others once at home and once away.(i.e. double round-robin schedule)
b) each team may participate in several away games in a week(from Sunday to Saturday).
c) there is no away game arrangement for a team, if it has a home game in the same week.
If the tournament finishes in 4 weeks, determine the maximum value of $n$.
1989 IMO Longlists, 7
For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.
1963 AMC 12/AHSME, 31
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 254\qquad
\textbf{(C)}\ 128 \qquad
\textbf{(D)}\ 127 \qquad
\textbf{(E)}\ 0$
2008 Gheorghe Vranceanu, 1
At what index the harmonic series has a fractional part of $ 1/12? $