Set Theory

Set theory is the foundational mathematical theory of sets, or collections of objects. It deals with operations on sets (union, intersection, complement, difference), properties of sets (cardinality, subsets), and relations between sets. Questions involve Venn diagrams, set notation, and properties.

• Question

  Consider the integers from `1` to `700`.

  a. How many of these numbers are even?

  b. How many of these numbers are divisible by `7`?

  c. How many of these numbers are not divisible by `2` nor by `7`?

  Answer question c.

  Topics:

  Number Theory -> Prime Numbers Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Matchings Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Set Theory

• Question

  In Alice's class, each student participates in at least one of the following clubs: English and Drama. In this class there are `30` students, `20` of whom study English, and `15` study Drama. How many students study in both clubs?

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  Topics:

  Set Theory

• Question

  Given `50` distinct natural numbers between `1` and `100`. It is known that no two of these numbers sum to `100`. Is it necessarily true that one of these numbers must be a perfect square?

  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Pigeonhole Principle Combinatorics -> Matchings Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  A wizard summoned `20` knights, of which `10` were elves and `10` were dwarves. The wizard wants to choose a team from them to perform a secret mission. This team must contain equal numbers of elves and dwarves.

  How many possibilities are there for such a team? (Note that he cannot choose an empty team)

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  Topics:

  Combinatorics -> Binomial Coefficients and Pascal's Triangle Set Theory

• More Numbers with Ascending Digits

  Miri writes down all the numbers whose decimal representation contains only the digits `1, 2, 3, 4, 5, 6`.
  (Not all digits must appear) and all the digits that appear are written in ascending order (e.g., 1356 or 124 or 5 but not 162 and not 1223).
  How many numbers will Miri write down?

  Topics:

  Combinatorics -> Product Rule / Rule of Product Set Theory
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 7

• We went on a trip

  Grade 4, consisting of 32 students, went on a trip. The students had to bring hats, sunglasses, and water bottles.
  No child forgot all of these things, but:

  • Among the students who brought hats, 9 forgot sunglasses,
  • Among the students who brought sunglasses, 7 forgot water bottles,
  • And among the students who brought water bottles, 10 forgot sunglasses.

  How many students in the class brought everything needed for the trip?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 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