Proof and Example, Constructing an Example / Counterexample

This involves finding a specific instance that satisfies a given set of conditions (an example) or one that disproves a general statement (a counterexample). It's a crucial skill for understanding mathematical claims. Questions directly ask for such constructions.

• 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