MathDB

5

Part of 2008 Iran MO (3rd Round)

Problems(5)

Irreducible in Z[x,y]

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Prove that the following polynomial is irreducible in Z[x,y] \mathbb Z[x,y]: x^{200}y^5\plus{}x^{51}y^{100}\plus{}x^{106}\minus{}4x^{100}y^5\plus{}x^{100}\minus{}2y^{100}\minus{}2x^6\plus{}4y^5\minus{}2
algebrapolynomialalgebra proposed
a+b+c|f(a)+f(b)+f(c)

Source: Iranian National Olympiad (3rd Round) 2008

8/31/2008
Find all polynomials f∈Z[x] f\in\mathbb Z[x] such that for each a,b,x∈N a,b,x\in\mathbb N a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)
algebrapolynomialmodular arithmeticfunctionnumber theory proposednumber theory
A game with ropes

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
n n people decide to play a game. There are n\minus{}1 ropes and each of its two ends are in hand of one of the players, in such a way that ropes and players form a tree. (Each person can hold more than rope end.) At each step a player gives one of the rope ends he is holding to another player. The goal is to make a path of length n\minus{}1 at the end. But the game regulations change before game starts. Everybody has to give one of his rope ends only two one of his neighbors. Let a a and b b be minimum steps for reaching to goal in these two games. Prove that a\equal{}b if and only if by removing all players with one rope end (leaves of the tree) the remaining people are on a path. (the remaining graph is a path.) http://i37.tinypic.com/2l9h1tv.png
combinatorics proposedcombinatorics
Feuerbach point

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let D,E,F D,E,F be tangency point of incircle of triangle ABC ABC with sides BC,AC,AB BC,AC,AB. DE DE and DF DF intersect the line from A A parallel to BC BC at K K and L L. Prove that the Euler line of triangle DKL DKL passes through Feuerbach point of triangle ABC ABC.
geometryEulergeometry proposed
The safe road

Source: Iranian National Olympiad (3rd Round) 2008

9/20/2008
a) Suppose that RBR′B′ RBR'B' is a convex quadrilateral such that vertices R R and R′ R' have red color and vertices B B and B′ B' have blue color. We put k k arbitrary points of colors blue and red in the quadrilateral such that no four of these k\plus{}4 point (except probably RBR′B′ RBR'B') lie one a circle. Prove that exactly one of the following cases occur? 1. There is a path from R R to R′ R' such that distance of every point on this path from one of red points is less than its distance from all blue points. 2. There is a path from B B to B′ B' such that distance of every point on this path from one of blue points is less than its distance from all red points. We call these two paths the blue path and the red path respectively. Let n n be a natural number. Two people play the following game. At each step one player puts a point in quadrilateral satisfying the above conditions. First player only puts red point and second player only puts blue points. Game finishes when every player has put n n points on the plane. First player's goal is to make a red path from R R to R′ R' and the second player's goal is to make a blue path from B B to B′ B'. b) Prove that if RBR′B′ RBR'B' is rectangle then for each n n the second player wins. c) Try to specify the winner for other quadrilaterals.
geometryrectanglegeometry proposed