Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9

• Question 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

• Question 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

• 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, Grades 7-8 Question 3

• Question 4 - Numbers Squared

  Ayala took all the numbers divisible by 3 from 1 to 99, inclusive, squared each of them, summed the results, and multiplied the resulting number by 2.
  The duck took all the numbers not divisible by 3 from 1 to 100, inclusive, squared each of them, and summed the results.
  What is the difference between the number the duck obtained and the number Ayala obtained?

  Topics:

  Arithmetic Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences

• Question 5 - How many solutions does the equation have?

  Given the equation:
  `x^2+2xy+y^2-200x-200y+1900=0`
  How many solutions (x,y) are there, where x and y
  are integers from 1 to 100 (inclusive)?

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations -> Diophantine Equations

• Question 6 - The Units Digit

  Miriam has eight cards with consecutive three-digit numbers. The units digit of the smallest number is 1,
  the units digit of the largest number is 8. Miriam arranged the cards in a row such that the first number is divisible by 2,
  the second number is divisible by 3, the third number is divisible by 4, and so on until the eighth number which is divisible by 9.
  What is the units digit of the number divisible by 7?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures