Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6

• Question 1

  A two-digit number is written on the board.
  Avi said: "The digit 5 appears in this number."
  Beni said: "This is a square number."
  Gili said: "This number is greater than 50."
  Dani said: "The number is divisible by 7."
  Then the teacher said: "There are three correct statements here and one incorrect statement.".
  What number was written on the board?

  Topics:

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

• Question 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

• Question 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
  Additional sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 4

• Question 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

• Question 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
  Additional sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 7

• Question 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

• Question 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