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.

• Question

  Is it possible for the sum of three natural numbers to be divisible by each of them?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  The following numbers are written on the board:

  `1,2^1,2^2,2^3,2^4,2^5`

  In one operation, you are allowed to erase two numbers written on the board and write their (non-negative) difference in their place.

  Is it possible to reach a state, through such operations, where only the number `15` is written on the board?

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  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences
  Sources:
  • problems.ru web site

• Question

  The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site

• The Sophisticated Task

  Hannah has a basket with `13` apples. Hannah wants to know the total weight of all these apples. Rachel has a digital scale, and she is willing to help Hannah, but only under the following conditions: In each weighing, Hannah can weigh exactly `2` apples, and the number of weighings cannot exceed `8`.

  Explain how, under these conditions, Hannah can know the total weight of the apples.

  Topics:

  Combinatorics -> Double Counting Algebra -> Equations Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Algorithm Theory -> Weighing
  Sources:
  • problems.ru web site

• Question

  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

  Given the polynomial `P(n)=n^2+n+41`. Is it true that this polynomial yields prime numbers for all natural numbers `n`?

  Topics:

  Number Theory -> Prime Numbers Algebra Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  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

  A grasshopper can jump `80` centimeters forward or `50` centimeters backward. Can the grasshopper move away from its starting point in fewer than `7` jumps to a distance of exactly one meter and `70` cm?

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  Topics:

  Arithmetic Number Theory -> Chinese Remainder Theorem Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations

• Question

  Is it possible to cut a triangle into four convex shapes: a triangle, a quadrilateral, a pentagon, and a hexagon?

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Question

  Given a sheet of paper of size `10×10` cm. Can you cut out a number of circles from this sheet such that the sum of their diameters is greater than `5` meters?

  Topics:

  Arithmetic Geometry -> Area Calculation Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems