Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10

• Question 1

  Find all integer solutions `(k>1) y^k=x^2+x`

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Algebra -> Equations -> Diophantine Equations

• Question 2

  Let M be a set of points in the plane. O is called a partial center of symmetry if it is possible to remove a point from M such that O is a regular center of symmetry of what remains. How many partial centers of symmetry can a finite set of points in the plane have? 
  V. Prasolov 

  Topics:

  Combinatorics -> Combinatorial Geometry Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Symmetry

• Question 3

  Let ABCD be a convex quadrilateral inscribed in a circle such that its diagonals are perpendicular to each other. Let O be the center of the circle. Prove that the broken line AOC divides the quadrilateral into two parts of equal area. 
  V. Varvarkin 

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Angle Calculation

• Question 4

  64 friends were told 64 news items at the same time (one news item per friend). They start calling each other and exchanging news. Each call lasts one hour. How long will it take for all the friends to know all the news? 
  א. אנג'אנס 

  Topics:

  Logic -> Reasoning / Logic

• Question 5 - Wolf and sheep

  The game takes place on an infinite plane. One player moves the wolf, and the other – 50 sheep. After a move by the wolf, one of the sheep makes a move, then the wolf again, and so on. In one move, the wolf or sheep moves no more than one meter in any direction. Can the wolf always catch at least one sheep, regardless of the initial configuration? 

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Combinatorics -> Game Theory Proof and Example -> Constructing an Example / Counterexample