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

• Question

  The numbers with a digit sum of 28 are written on the board in ascending order. What is the 24th number among them?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 2

• Circle of Liars - The Truth Claim

  In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.

  The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.

  Each of the people knows exactly who is a liar and who is a truth-teller.

  Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.

  It is known that in the circle there is at least one liar, and at least one truth-teller.

  a. Is it possible that `n = 2017`?

  b. Is it possible that `n = 5778`?

  (Solution format: "word, word" for example "cat, puppy")

  Show hint

  Log in here to receive the hint
  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems
  Sources:
  • Gillis Mathematical Olympiad, 2017-2018 Question 1

• A 2022x2022 Board and Inversion Operations

  We have a `2022 times 2022` board with real numbers.

  In each move, we can choose a row or a column and a real number `c`.

  Then, we replace each number in the row or column from `x` to `c - x`.

  Is it possible to get from any board to any other board?

   

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics -> Invariants Algebra Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables

• Number Network

  In the diagram, the numbers on the edges indicate the differences between the numbers inside the circles. Place positive numbers inside the circles and discover what the number in the bottom circle is.

  

  Topics:

  Arithmetic Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 6

• Letter Replacement

  Each letter represents a different digit; whenever a specific letter appears, it is necessarily the same digit.

  Find `B-E/2`

  Given: `AB*C=DE`

  And also `F^D=GF`

  Show hint

  Log in here to receive the hint
  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Squares

  You have many cardboard squares of sizes `1 times 1`, `2 times 2`, and `3 times 3`, and you must assemble them into a square of size `7 times 7`.
  What is the smallest possible number of squares you will need?

  Topics:

  Geometry Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 5

• Question

  A two-digit number is written on the board.
  Avi said: "The digit 5 appears in this number."
  Beni said: "This is a square number."
  Gili said: "This number is greater than 50."
  Dani said: "The number is divisible by 7."
  Then the teacher said: "There are three correct statements here and one incorrect statement.".
  What number was written on the board?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 1

• Divisible by 2 or 5 but not 3

  How many five-digit numbers are divisible by 2 or 5, but not divisible by 3?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Product Rule / Rule of Product
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 4

• Minimal Table

  Given a table of size `3 times 3`. Hilla wants to write digits from 1 to 9 in the table's cells, such that all the sums in the rows and columns of the table are different, and the total sum of the table is as small as possible.
  It is allowed to repeat the same digit multiple times. What is the smallest sum that Hilla can obtain?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 7

• How Many Liars?

  A tourist is traveling in a land of liars and truth-tellers. All truth-tellers always tell the truth, and all liars always lie.
  The tourist meets four friends: Alice, Betty, John, and Donald, and asks them: "How many of the four of you are liars?"
  Alice answers: 0
  Betty answers: 1
  John answers: 2
  Donald answers: 3
  Can we know for certain how many of them are liars?

   

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 1