Found problems: 259
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?
The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.
$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?
$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.
Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened.
Decide whether there exists a strategy for player $A$ to win in a finite number of moves.
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$.
A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?
[i]Proposed by Michael Kural[/i]
Cells of a $2000\times2000$ board are colored according to the following rules:
1)At any moment a cell can be colored, if none of its neighbors are colored
2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored.
3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored
(Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored?
[I]Proposed by K. Kohas[/i]
A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$
A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle.
[color=red][Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.][/color]
The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?
In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).
Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
Petya chooses $100$ pairwise distinct positive numbers less than $1$ and arranges them in a circle. In one operation, he may take three consecutive numbers \( a, b, c \) (in this order) and replace \( b \) with \( a - b + c \). What is the greatest value of \( k \) such that Petya could initially choose the numbers and perform several operations so that \( k \) of the resulting numbers are integers? \\
Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the integers $x$ and $y$, where $x\leq y$ and replace them with $2x$ and $y-x.$
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that:
(i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again,
(ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps,
(iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.
Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers.
$(a)$ Find the minimal possible value of this sum.
$(b)$ Find the maximal possible value of this sum.
Five numbers 1,2,3,4,5 are written on a blackboard. A student may
erase any two of the numbers a and b on the board and write the
numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?
Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]