Proof and Example

This category emphasizes the core mathematical activities of constructing rigorous arguments (proofs) to establish general truths, and using specific instances (examples) to illustrate concepts, test conjectures, or find counterexamples. Questions may ask for either or both.

• Question

  The numbers `1,2,3,4,5` are written at the vertices of a regular pentagon, with each number at exactly one vertex. A trio of vertices is called successful if it forms an isosceles triangle, where the number at its apex is greater than the numbers at the other two vertices, or where the number at its apex is smaller than the numbers at the other two vertices.

  Find the maximum number of successful trios that can exist.

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Symmetry Minimum and Maximum Problems / Optimization Problems

• Ten-Digit Number

  Yael writes ten-digit numbers in whose decimal representation each of the digits `0, 1, 2, 3, 4, 5, 6, 7, 8, 9` appears exactly once.
  In the numbers that Yael writes, the difference between any two adjacent digits is at least 2. What is the smallest number Yael can write?
   

  Topics:

  Number Theory Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 3
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 1

• The Round Table

  Around a round table are 12 chairs, with knights sitting on some of them. Arthur wants to join the meeting,
  and it turns out that no matter where he sits, someone is definitely sitting next to him.
  What is the smallest number of knights that can be around the table to ensure this is true? (not including Arthur)

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 1

• Colorful Street 2

  There are 15 houses along the street, colored red, blue, and green. There is at least one house of each color.
  Between any two blue houses there is a red house. Between any two green houses there is a blue house.
  What is the largest possible number of green houses?
  Note: The street is straight, and all houses are located on one side of the street.

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 5
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 6

• Integer Coefficients?

  Given real numbers a, b, c such that for every integer x, the number `ax^2+bx+c` is an integer. Does this necessarily imply that a, b, c are all integers? Prove it, or provide a counterexample.

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  Topics:

  Algebra Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • Grossman Math Olympiad, 2017, Juniors Question 2

• Sets in the Plane

  A. Does there exist a set A in the plane such that its intersection with every circle contains exactly two points?

  B. Does there exist a set B in the plane such that its intersection with every circle of radius 1 contains exactly two points?

  Topics:

  Geometry -> Plane Geometry -> Circles Proof and Example -> Constructing an Example / Counterexample Set Theory Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Grossman Math Olympiad, 2006 Question 3

• The 1224th Digit

  We write the natural numbers in order, one after the other from left to right:

  1234567891011...

  Note, for example, that the digit in the 10th place is 1 and the digit in the 11th place is 0.

  Continuing with this writing as much as needed...

  Which digit will be in the 1224th place in the sequence?

  Topics:

  Number Theory Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 1

• It's Crowded Here!

  55 gears are placed on the game board in the shape of a 'pyramid':

  10 gears in the bottom row, 9 gears in the row above, and so on.

  In this state, the gears cannot rotate freely (convince yourself why!)

  Remove gears to allow free movement.

  What is the maximum number of gears that can remain on the board so that they can all rotate?

  

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 10

• Paths in a Triangular Park

  In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 3

• Question

  An `N×N` table is filled with numbers such that all rows are distinct (differing in at least one position). Prove that it is possible to delete a column such that in the remaining table all rows are also distinct.
  (a) Hint

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Graph Theory Proof and Example -> Proof by Contradiction
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 2