Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8

• Question 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

• Question 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

• Question 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
  Additional sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 3

• Question 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

• Question 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

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