Combinatorics

Combinatorics is the art of counting. It deals with selections, arrangements, and combinations of objects. Questions involve determining the number of ways to perform tasks, arrange items (permutations), or choose subsets (combinations), often using principles like the product rule and sum rule.

Pigeonhole Principle Double Counting Binomial Coefficients and Pascal's Triangle Product Rule / Rule of Product Graph Theory Matchings Induction (Mathematical Induction) Game Theory Combinatorial Geometry Invariants Case Analysis / Checking Cases Processes / Procedures Number Tables Colorings

• Who Likes Cola?

  Shmuel has 4 children: Ohad, Benny, Guy, and Dor. Each of them has one drink that he likes (no two people share a favorite drink): water, cola, grape juice, and orange juice.
  In addition, each of them also has a favorite subject (and no two have the same one): mathematics, physics, computer science, and chemistry.

  Hints:

  1. 1. Ohad likes computer science
  2. 2. The one who likes mathematics does not drink orange juice
  3. 3. Dor does not like physics
  4. 4. Benny drinks water
  5. 5. Guy does not drink grape juice
  6. 6. The child who likes chemistry drinks grape juice
  7. 7. Benny does not like mathematics

  Question: Who likes cola?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 2

• Dissect into four parts

  Geometric shapes are called congruent if they coincide when superimposed. Cut the following shape into four congruent parts:

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 3

• Compose a Sum

  Compose three numbers from the digits `1,2,3,4,5,6,7,8,9` such that one of them is the sum of the other two.
  Each digit must be used exactly once.

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 1

• How Many Sums Are Possible?

  What is the number of possible results of addition exercises of the form `A + B`
  where A and B are two distinct positive single-digit numbers?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 5

• 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

• Hunter in the Enchanted Forest

  In the enchanted forest lives a tribe of 150 people. Every day, the tribe members try to organize themselves to go hunting together,
  and each day, each person participates or does not participate according to their choice. Prove that during one week, there will certainly be two people who arrive on exactly the same days.
  Note: Even in the enchanted forest, a week has 7 days.

  Topics:

  Combinatorics -> Pigeonhole Principle
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 4

• Cut a Boat in Half

  Geometric shapes are called congruent if they coincide when placed on top of each other. Cut the shape into two congruent parts

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 6

• The missing number

  Find a number such that if we write it in each of the blue squares, the sums in all `3times3` squares in the table will be equal to each other.

  

  Topics:

  Combinatorics -> Number Tables
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 1

• Five-Digit Palindromic Numbers

  A number is called palindromic if its digits can be read in reverse order, and the result will not change. For example, the number 4407667044 is a palindromic number, and so is 92329.
  How many five-digit palindromic numbers are there in total?

  Topics:

  Arithmetic Combinatorics
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 2

• Largest Perimeter

  A polygon with an area of 12 is drawn on grid paper, with all its sides passing through the grid lines. What is the largest possible perimeter of this polygon?

  Topics:

  Geometry -> Area Calculation Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 5