Found problems: 1187
2014 Belarus Team Selection Test, 3
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
2018 Stars of Mathematics, 3
Given a positive integer $n$, determine the largest integer $M$ satisfying
$$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$
for all non-negative integers $a_1,...., a_n$.
S. Berlov, A. Khrabrov
1990 IMO Longlists, 29
Function $f(n), n \in \mathbb N$, is defined as follows:
Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$
2011 Romania Team Selection Test, 2
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]
1982 AMC 12/AHSME, 15
Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations
\[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is
$\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$
2014 Contests, 1
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
2021 Brazil National Olympiad, 3
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 \(n>N\).
2023 Harvard-MIT Mathematics Tournament, 8
Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]
2010 AMC 12/AHSME, 24
Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2003 Estonia National Olympiad, 4
Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.
2012 Romanian Masters In Mathematics, 5
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]