MathDB

2009 numbers on a circle

Source: All Russian Olympiad 2009 - Problem 10.4

5/15/2009

On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most

k

times. What is the minimum

k

so that we can make all the numbers on the circle equal?

modular arithmeticgeometrygeometric transformationrotationcombinatorics unsolvedcombinatorics

Nice problem about finding a coin

Source: ARO 2009, Round 5, 9 grade, Problem 4

5/3/2009

There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?

modular arithmeticcombinatoricsRussia

A game with a set of points

Source: All Russian Olympiad 2009 - Problem 11.4

5/17/2009

Given a set

M

of points

(x,y)

with integral coordinates satisfying

x^2 + y^2\leq 10^{10}

. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

calculusintegrationanalytic geometrygeometrygeometric transformationreflectioncombinatorics proposed

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Problems(3)