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.

• 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.

  Show hint

  Log in here to receive the hint
  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

• 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?

  Show hint

  Log in here to receive the hint
  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?

  Show hint

  Log in here to receive the hint
  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