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

  Given three real numbers `a`, `b`, and `c`. It is known that `a+b+c>0`, `ab+bc+ca>0`, and `abc>0`. Prove that `a>0`, `b>0`, and `c>0`.

  Topics:

  Algebra -> Inequalities Proof and Example -> Proof by Contradiction

• Question

  The magical land consists of `25` provinces. Is it possible that each province borders an odd number of other provinces?

  Topics:

  Combinatorics -> Double Counting Combinatorics -> Graph Theory Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings

• Question

  Prove that there is no polyhedron with `7` edges.

  Topics:

  Combinatorics -> Graph Theory Proof and Example -> Proof by Contradiction Geometry -> Solid Geometry / Geometry in Space -> Polyhedra

• Question

  Given seven integers `a_1,a_2,a_3,...,a_7`, and let `b_1,b_2,b_3,...,b_7` be the same numbers written in a different order. Prove that the number `(a_1-b_1)(a_2-b_2)*...*(a_7-b_7)` must be even.

  Topics:

  Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  From a chessboard, two opposite corners are removed (the squares `a1` and `h8`, for example). Can you tile the remaining board with dominoes?

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  Topics:

  Combinatorics -> Invariants Combinatorics -> Matchings Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring Puzzles and Rebuses

• Knights of the Round Table

  Around a round table sit 12 knights, each of whom is either an elf or a dwarf. It is known that the number of elves is greater than the number of dwarves. Prove that there are two elves who sit opposite each other.

  Will this continue to be true if the total number of knights is 120?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction

• Question

  A total of `21` children have `200` nuts. Prove that there exist two children who have the same number of nuts.

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Question

  There are `30` students in a class. During a test, Pinchas made `13` mistakes, and the rest made fewer mistakes. Prove that there are three students who made the same number of mistakes.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Question

  The numbers from `1` to `2n` are written in a row in some order. Add to each number the index of the position it stands on. Prove that among the `2n` sums we obtained, there are two whose difference is divisible by `2n`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Combinatorics -> Pigeonhole Principle Proof and Example -> Proof by Contradiction

• Question

  A square is divided into several convex polygons (more than `1`), each of which has a different number of sides. Prove that among these polygons there is a triangle.

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Combinatorics -> Graph Theory Geometry -> Plane Geometry -> Triangles Proof and Example -> Proof by Contradiction Geometry -> Solid Geometry / Geometry in Space -> Polyhedra Minimum and Maximum Problems / Optimization Problems