Young Mathematician Olympiad, 2018-2019

Stage A Stage B Final

• Question from sources: Stage B, Grades 5-6(2) - ABC

  In the following exercise, different digits have been replaced with different letters, and identical digits have been replaced with identical digits. Find the three-digit number ABC

  `ABC - A -B-C=DCA`

  Topics:

  Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 2

• Question from sources: Stage B, Grades 3-4(3), Stage B, Grades 5-6(3) - Candies

  In a class, there are a number of students and each has a number of candies:
  There are exactly 10 children with at least one candy,
  Exactly 8 children with at least two candies,
  Exactly 6 children with at least 3,
  Exactly 4 children with at least 4,
  And exactly 2 children with 5 candies.
  It is known that no one has more than 5 candies. How many candies are there in the class?

  Topics:

  Arithmetic Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 3

• Question from sources: Stage B, Grades 5-6(4) - Divisible by 2 or 5 but not 3

  How many five-digit numbers are divisible by 2 or 5, but not divisible by 3?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Product Rule / Rule of Product
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 4

• Question from sources: Stage B, Grades 3-4(5), Stage B, Grades 5-6(5) - Hexagon and Triangle

  A regular hexagon and an equilateral triangle have the same perimeter. The area of the triangle is known to be 60. Find the area of the hexagon.

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 5

• Question from sources: Stage B, Grades 5-6(6) - Horse, Camel, and Donkey in a Circle

  On a circular track of length 92, there is a horse, a donkey, and a camel that start from the same point and begin walking along the circle.
  The horse and the camel walk counterclockwise, and the donkey walks clockwise. The camel's speed is 1 meter per second, the donkey's is 3, and the horse's is 5.

  In how many seconds will all three meet again?

  Note: The meeting does not necessarily have to be at the starting point

  Topics:

  Algebra -> Word Problems -> Motion Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 6

• Question from sources: Stage B, Grades 5-6(7) - Minimal Table

  Given a table of size `3 times 3`. Hilla wants to write digits from 1 to 9 in the table's cells, such that all the sums in the rows and columns of the table are different, and the total sum of the table is as small as possible.
  It is allowed to repeat the same digit multiple times. What is the smallest sum that Hilla can obtain?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 7

• Question from sources: Final, Grades 3-4(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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 1

• Question from sources: Final, Grades 3-4(2), Final, Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 1

• Question from sources: Final, Grades 3-4(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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 3

• Question from sources: Final, Grades 3-4(4), Final, Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 2