Young Mathematician Olympiad, 2020-2021, Stage A

Grades 3-4 Grades 5-6 Grades 7-8 Grade 9

• Question from sources: Grades 5-6(5), Grades 7-8(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
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 5
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 6

• Question from sources: Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 6

• Question from sources: Grades 7-8(1) - Average of Averages 2

  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?

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 1

• Question from sources: Grades 7-8(2) - Length of the Segment

  On the side BC of triangle ABC, a point D is marked. The perimeter of triangle ABC is 15 centimeters, the perimeter of triangle ABD is 12 centimeters, and the perimeter of triangle ACD is 13 centimeters.
  What is the length of segment AD?

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 2

• Question from sources: Grades 7-8(3), Grade 9(3) - Clock Angle

  How many minutes after 7:00 will the angle between the hour and minute hands be exactly one degree for the first time?
  Note: The clock hands move continuously at a constant speed.

  Topics:

  Arithmetic Algebra -> Word Problems -> Motion Problems Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 3
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 3

• Question from sources: Grades 7-8(4) - Star of David in a Circle

  Eight points are given on a circle. In how many different ways can a Star of David be drawn with vertices at these points?

  

  Note: A Star of David is a figure obtained when two triangles intersect and their sides create exactly 6 intersection points.

  Topics:

  Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 4

• Question from sources: Grades 7-8(5) - Consecutive Numbers and Deletion

  Yossi wrote 10 consecutive natural numbers on the board. Danny erased one of the numbers. The sum of the remaining numbers on the board is 2020.
  Which number was erased?

  Topics:

  Algebra -> Word Problems Algebra -> Sequences Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 5

• Question from sources: Grades 5-6(6), Grades 7-8(6) - 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
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 5
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 6

• Question from sources: Grade 9(1) - Average of Averages 3

  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?

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 1

• Question from sources: Grade 9(2) - Area of the Shape

  In the image, the area of the semicircle is equal to 1. Find the area of the larger shape, given that all the curved lines in the image are quarter circles.

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 2