Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6

• Question 1 - Average of Averages

  Consider the following diagram:

  

  Each number in the diagram, connected to 4 other numbers, must be equal to their average:
  What is the number in the circle marked with a question mark?
  Note: The average of four numbers is their sum divided by 4.

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means

• Question 2 - The Truth-Teller and Liar Survey

  13 truth-tellers and 12 liars participated in a survey. In the survey, each participant was asked about every other participant (including themselves) whether they were a truth-teller. How many "yes" answers were received in the survey in total?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems

• Question 3 - The Orange Path

  In the illustration, there is an orange path surrounding a blue square. The area of the path is 44% of the area of the square.
  What is the width of the orange path as a percentage relative to the side length of the blue square?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems Arithmetic -> Percentages

• Question 4 - Pairwise Relatively Prime Composite Numbers

  Yossi writes two-digit composite numbers on the board. He wants all the numbers written on the board to be pairwise relatively prime.
  What is the maximum number of integers Yossi can write on the board?
  Note: Integers are called relatively prime if they have no common factors other than 1.

  Topics:

  Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

• Question 5 - Colorful Street 2

  There are 15 houses along the street, colored red, blue, and green. There is at least one house of each color.
  Between any two blue houses there is a red house. Between any two green houses there is a blue house.
  What is the largest possible number of green houses?
  Note: The street is straight, and all houses are located on one side of the street.

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Additional sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 6

• Question 6 - Magic Fractions

  Let's call a fraction magic if both its numerator and denominator are less than 10. For example, the fraction `1/9` is considered magic, the fraction `6/8` is also magic, but the fraction `3/14` is not magic.
  How many magic fractions are there that are greater than one-half and less than 1?

  Note: For the purpose of this question, `2/3` and `4/6` are considered different fractions.

  Topics:

  Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures