Tournament of Towns, 1980-1981

• Question from sources: Spring, Main Version, Grades 9-10(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
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 1 Points 3

• Question from sources: Spring, Main Version, Grades 9-10(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
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 2 Points 7

• Question from sources: Spring, Main Version, Grades 9-10(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
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 3 Points 5

• Question from sources: Spring, Main Version, Grades 9-10(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
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 4 Points 8

• Question from sources: Spring, Main Version, Grades 9-10(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
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 5 Points 16

• Question from sources: Spring, Main Version, Grades 11-12(1)

  Suppose two pyramids are tangent to each other if they have no common interior points and they intersect in a non-degenerate planar polygon. Is it possible for 8 pyramids in space to all be tangent to each other? 
  A. Angeles 

  Topics:

  Combinatorics -> Combinatorial Geometry Proof and Example -> Constructing an Example / Counterexample Geometry -> Solid Geometry / Geometry in Space -> Polyhedra
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 1 Points 7

• Question from sources: Spring, Main Version, Grades 11-12(2)

  The game takes place on an infinite plane. One player moves the wolf, and another player moves K sheep. After the wolf's move, one of the sheep makes a move, then the wolf again, and so on. In one move, the wolf or a sheep cannot move more than one meter in any direction. Can the wolf always catch at least one sheep, regardless of the initial positions?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Geometry -> Plane Geometry Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 2

• Question from sources: Spring, Main Version, Grades 11-12(3)

  Prove that any real number can be written as the sum of 9 numbers, each of which is composed only of the digits 0 and 7.

  Topics:

  Number Theory Arithmetic Proof and Example Algebra -> Sequences
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 3

• Question from sources: Spring, Main Version, Grades 11-12(4)

  K friends simultaneously learn K pieces of news (one piece of news per friend). They begin to phone each other and exchange news. Each call lasts one hour. How long will it take for all friends to know all the news? Consider the cases:
  (a) (5 points) K=64
  (b) (10 points) K=55
  (c) (12 points) K=100
  (a) Answer

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 4

• Question from sources: Spring, Main Version, Grades 11-12(5)

  On an infinite grid of squares, 6 squares are marked, as in the diagram. How many squares contain stones? In one move, a stone can be removed if it has no adjacent stone above and to its right; then, two stones are placed in the squares above and to the right of the removed stone. Can we remove all stones from the marked squares if the initial state is:
  A. (8 points) All marked squares.
  B. (8 points) Only the bottom-left marked square.

  O 
  O O 
  O O O

  M. Konvitz'

  Topics:

  Combinatorics -> Invariants Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 5