Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4

• Question 1 - 5 Lines, 8 Intersections

  Draw 5 lines such that there are exactly 8 points of intersection between them.

  Topics:

  Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 2 - 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

• Question 3 - 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)

• Question 4 - 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
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 1

• Question 5 - 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

• Question 6 - Cards with Digits

  Rachel has three cards with different digits, all of which are greater than 0. Rachel formed all possible three-digit numbers from these cards and calculated their sum.
  Prove that the sum is divisible by 3

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Proof and Example
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 5