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

  What is the smallest six-digit number with all different digits?

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  Topics:

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

• Question

  Three hedgehogs have three pieces of cheese weighing `5`, `8`, and `11` grams. A fox offers to help the hedgehogs divide the cheese equally. The fox can bite off one gram from each of two cheese pieces of its choice. Can the fox, using these actions, reach a state where it leaves the three hedgehogs with equal pieces of cheese?

  Topics:

  Arithmetic Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Can you divide `24` kilograms of nails into two parts of `15` and `9` kilograms using a balance scale without weights?

  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algorithm Theory -> Weighing

• The broken chain

  There are five segments of a broken chain, each segment having three links. Moses wants to repair the chain. What is the minimum number of links he needs to open and close again to join all these segments together?

  Note: The chain is not circular!

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems

• Question

  Is it possible to arrange all the numbers from `1` to `100` in a row such that the difference between any two adjacent numbers is at least `50`? If so, provide an example; if not, prove why.

  Topics:

  Number Theory Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  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

• 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

• Magic Number 15

  Yossi writes the number `15` on the board. Then, Danny adds a digit to the right and a digit to the left of the number written on the board, such that the new number is still divisible by `15`.

  Find this number. Is there only one possibility?

  Note: The digit added to the left is not zero.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Constructing an Example / Counterexample Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Find five natural numbers whose sum is `20`, and whose product is `420`.

  Topics:

  Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Question

  The number `458` is written on the board. In each single step, you are allowed to either multiply the number written on the board by `2`, or erase its last digit.

  Is it possible to obtain the number `14` using these operations?

  Topics:

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