Found problems: 1187
Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?
Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.
Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.
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
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$
[i](8 points)[/i]
Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $
[i]Petre Guțescu[/i]
Prove the following equalities of sets:
\[ \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right)\]
\[ \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)\]
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]
For every integer $n \ge 1$, the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$. Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$. (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
[i]Proposed by Lewis Chen[/i]
(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$.
Calculate $[M(a,b)]$
(ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that
i) $|A_i|=3$ for each $i=1,\ldots ,m$.
ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.
Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?
$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$.
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is [i]tough[/i] when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$.
Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.
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 $T^n(J) \cap J \neq \emptyset.$
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that
\[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\]
and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.
Find
\[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\]
[i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]
What is the largest integer less than or equal to $$\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?$$
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
[i]Proposed by Evan Chen[/i]