Found problems: 815
Greedy goblin Griphook has a regular $2000$-gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins he could have collected?
[i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]
We call the arrangement of $n$ ones and $m$ zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural $m$ and $n$ does a good arrangement exist?
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]
Let $n\ge3$ be a positive integer. Alex and Lizzie play a game. Alex chooses $n$ positive integers (not necessarily distinct), writes them on a blackboard, and does nothing further. Then, Lizzie is allowed to pick some of the numbers―but not all of them―and replace them each by their average. For example, if $n=7$ and the numbers Alex writes on the blackboard to start are $1,2,4,5,9,4,11$, then on her first turn Lizzie could pick $1,4,9$, erase them, and replace them each with the number $\tfrac{1+4+9}{3}$, leaving on the blackboard the numbers $\tfrac{14}{3},2,\tfrac{14}{3},5,\tfrac{14}{3},4,11$. Lizzie can repeat this process of selecting and averaging some numbers as often as she wants. Lizzie wins the game if eventually all of the numbers written on the blackboard are equal. Find all positive integers $n\ge3$ such that no matter what numbers Alex picks, Lizzie can win the game.
Let $m$ be a positive integers. A square room with corners at $(0,0), (2m,0), (0,2m),$ $(2m,2m)$ has mirrors as walls. At each integer lattice point $(i,j)$ with $0 < i, j < 2m$ a single small double sided mirror is oriented parallel to either the $x$ or $y$ axis. A beam of light is shone from a corner making a $45^\circ$ angle with each of the walls. Prove that the opposite corner is not lit.
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.$
The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$?
[i] (Andrei Bâra)[/i]
There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$.
[i]Proposed by Dusan Dukic, Serbia[/i]
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
[i]Proposed by Jorge Tipe, Peru[/i]
Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$.
[i](4 points)[/i]
Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform
($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $
into
$ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $
by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.
[i]Andrew Gu[/i]
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.
There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$.
[i]Proposed by Dusan Dukic, Serbia[/i]
Fifteen stones are placed on a $4 \times 4$ board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.
For which initial positions of the empty cell is it possible to end up with exactly one stone on the board?
At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. Call a person [i]junior[/i] if they are younger than all their friends, and [i]senior[/i] if they are older than all their friends. A person with no friends is both [i]junior[/i] and [i]senior[/i]. A sequence of pairwise distinct people $A_1, \dots, A_m$ is called [i]photogenic[/i] if:
1. $A_1$ is [i]junior[/i],
2. $A_m$ is [i]senior[/i], and
3. $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \leq i \leq m-1$.
Let $k$ be a positive integer such that for every [i]photogenic[/i] sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends.
[i]Proposed by Shantanu Nene[/i]
Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are $20$ cookies on a red plate and $14$ on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
A regular $ (5 \times 5)$-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.
A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations:
[b]i)[/b] to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile;
[b]ii)[/b] to choose a pile with at least $4$ stones, remove $2$ stones from it, and then split it into two piles with amount of piles to be chosen by the player.
The game continues until is not possible to make an operation. Show that the number of piles with one stone in the end of the game is always the same, no matter how the movements are made.
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$
[i]Author: Gerhard Wöginger, Netherlands[/i]
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.
Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.
Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that \[ \left\lvert x\right\rvert + \left\lvert y + \frac{1}{2} \right\rvert < n. \] A path is a sequence of distinct points $(x_1 , y_1), (x_2, y_2), \ldots, (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \ldots, \ell$, the distance between $(x_i , y_i)$ and $(x_{i-1} , y_{i-1} )$ is $1$ (in other words, the points $(x_i, y_i)$ and $(x_{i-1} , y_{i-1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).
Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.