Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4

• Question 1 - How Many Liars?

  A tourist is traveling in a land of liars and truth-tellers. All truth-tellers always tell the truth, and all liars always lie.
  The tourist meets four friends: Alice, Betty, John, and Donald, and asks them: "How many of the four of you are liars?"
  Alice answers: 0
  Betty answers: 1
  John answers: 2
  Donald answers: 3
  Can we know for certain how many of them are liars?

   

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems

• Question 2 - Parentheses

  Add parentheses to make the result as large as possible:

  `10000-1000-100-10-1`

  Topics:

  Arithmetic Algebra -> Equations Minimum and Maximum Problems / Optimization Problems
  Additional sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 1

• Question 3 - Donkey in the Middle

  Avi, Beni, and Gadi played "Donkey in the Middle" - at any moment someone stands in the middle and tries to catch a ball that the other two are passing. If he succeeds, one of the other two replaces him.
  After the game, it turned out that Avi stood in the middle 8 times, Beni 4 times, and Gadi 13 times.
  Who was the first and who was the last to stand in the middle?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 4 - The Beaver and the Mole

  There is a plot of land in the shape of a square `4 times 4` divided into cells of `1 times 1`. The beaver wants to build a house on it that occupies 4 cells, which from a top-down view looks like this:

  

  The mole wants to disturb him. For this purpose, it can dig holes, each of which occupies one cell. It is impossible to build on the cells that have become holes. What is the smallest number of holes the mole needs to dig so that the beaver cannot build the house?

   

  Topics:

  Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Additional sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 2

• Question 5 - Doughnuts

  Ayala, Benny, Gili, Danny, and Hadas received a package of doughnuts containing

  • 10 doughnuts with dulce de leche
  • 8 doughnuts with peanut butter
  • 9 doughnuts with chocolate
  • 11 with strawberry jam

  Each of them has their favorite type of doughnut.

  • Ayala ate 5 doughnuts of her favorite type
  • Benny ate 6 doughnuts of his favorite type
  • Gili ate 7 doughnuts of her favorite type
  • Danny ate 8 doughnuts of his favorite type
  • Hadas ate 9 doughnuts of her favorite type

  After that, they were left with 3 doughnuts of different types. What is each of their favorite type of doughnut?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 6 - 6 on the Board

  The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
  Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no more

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Additional sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 5

• Question 7 - Sum of Squares

  Given a rectangle with an area of 13 and a perimeter of 20. On two adjacent sides of the rectangle, two squares are constructed, as shown in the figure. Find the sum of the areas of the squares.

  

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  Topics:

  Geometry -> Area Calculation Algebra -> Equations Algebra -> Word Problems