Found problems: 1187
(a) We say that a hyperplane $H$ that is given with this equation
\[H=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n=b\}\]
($a=(a_1,\dots,a_n)\in \mathbb R^n$ and $b\in \mathbb R$ constant) bisects the finite set $A\subseteq \mathbb R^n$ if each of the two halfspaces $H^+=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n>b\}$ and $H^-=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n<b\}$ have at most $\lfloor \tfrac{|A|}{2}\rfloor$ points of $A$.
Suppose that $A_1,\dots,A_n$ are finite subsets of $\mathbb R^n$. Prove that there exists a hyperplane $H$ in $\mathbb R^n$ that bisects all of them at the same time.
(b) Suppose that the points in $B=A_1\cup \dots \cup A_n$ are in general position. Prove that there exists a hyperplane $H$ such that $H^+\cap A_i$ and $H^-\cap A_i$ contain exactly $\lfloor \tfrac{|A_i|}{2}\rfloor$ points of $A_i$.
(c) With the help of part (b), show that the following theorem is true: Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even, such that each of the robbers receive the same number of stones of each kind. Show that the two robbers can accomplish this by cutting the necklace in at most $d$ places.
[i]The Algorithm.[/i] There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane:
\[\begin{array}{ccc}A=(1,10)&B=(976,9)&C=(666,87)\\D=(377,422)&E=(535,488)&F=(775,488) \\ G=(941,500) & H=(225,583)&I=(388,696)\\J=(3,713)&K=(504,872)&L=(560,934)\\&M=(22,997)&\end{array}\]
At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the [i]total downtime[/i] of the computers? List the points he visits in order. Your score will be $\left\lfloor \dfrac{N}{40}\right\rfloor$, where \[N=1000+\lfloor\text{the optimal downtime}\rfloor - \lfloor \text{your downtime}\rfloor ,\] or $0$, whichever is greater. By total downtime we mean the sum \[\sum_{P\in S}t_P,\] where $t_P$ is the time at which the repairman reaches $P$.
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
Compute the sum of all positive real numbers $x \le 5$ satisfying $$x =\frac{ \lceil x^2 \rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{ \lceil x\rceil + \lfloor x \rfloor}$$
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.
If $n\ge2$ is an integer, prove the equality
$$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$
[color=darkblue]Weightlifter Ruslan has just finished the exercise with a weight, which has $ n$ small weights on one side and $ n$ on the another. At each stage he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $ k$. What is the minimal number of stages (in function if $ n$ and $ k$), which Ruslan need to take off all weights..[/color]
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$).
b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
Let $n$ and $k$ are integers with $n>0$. Prove that \[-\frac{1}{2n}\sum^{n-1}_{m=1}\cot \frac{\pi m}{n}\sin \frac{2\pi km}{n}= \begin{cases}\tfrac{k}{n}-\lfloor\tfrac{k}{n}\rfloor-\frac12 & \text{if }k|n \\ 0 & \text{otherwise}\end{cases}.\]
Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.
$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function
[list=1]
[*] -1
[*] 0
[*] 1
[*] Does not exists
[/list]
A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that:
1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$.
2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\]
a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have:
\[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\]
b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.
Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 254\qquad
\textbf{(C)}\ 128 \qquad
\textbf{(D)}\ 127 \qquad
\textbf{(E)}\ 0$
Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$
Find all integer solutions of the equation \[\left\lfloor \frac{x}{1!}\right\rfloor+\left\lfloor \frac{x}{2!}\right\rfloor+\cdots+\left\lfloor \frac{x}{10!}\right\rfloor =1001.\]
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.
Let be a natural number $ n\ge 2. $ Prove that there exists an unique bipartition $ \left( A,B \right) $ of the set $ \{ 1,2\ldots ,n \} $ such that $ \lfloor \sqrt x \rfloor\neq y , $ for any $ x,y\in A , $ and $ \lfloor \sqrt z \rfloor\neq t , $ for any $ z,t\in B. $
[i]Costin Bădică[/i]
On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules:
$-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$;
$-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$;
$-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner.
Find which player has a winning strategy in each of the following cases:
a) $n_0=120$;
b) $n_0=\dsp \frac {3^{2002}-1}2$;
c) $n_0=\dsp \frac{3^{2002}+1}2$.
A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.
Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that
a) the sequence contains an even term;
b) the sequence contains infinitely many even terms.
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.