Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6

• Question 1 - Sheep and Camels

  Shmuel has a flock of 9 sheep and 5 camels. He wants to divide the flock between his two sons, Yossi and Danny, so that each of them receives an equal share of the value.
  It is known that 7 sheep cost as much as 3 camels. How can Shmuel divide the flock between his two sons equally, without selling any animals?

  Topics:

  Algebra -> Equations Algebra -> Word Problems Arithmetic -> Fractions
  Additional sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 5

• Question 2 - Log of Wood

  You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
  а. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
  б. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,

  Assuming you have no other measuring tools? Explain!

  Topics:

  Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder
  Additional sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 7

• Question 3 - Product of Areas

  In the diagram, there is a quadrilateral with perpendicular diagonals. Prove that the product of the areas of the shaded regions within the quadrilateral is equal to the product of the areas of the unshaded regions within the quadrilateral.

  

  Show hint

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  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles

• Question 4 - The Magic Octopuses

  In the magic sea live octopuses who can talk. Each octopus either always tells the truth or always lies. One day
  the following conversation took place between four octopuses, Avi, Benny, Gidi, and Danny:

  
  Avi: I am a green octopus
  Benny: I am not green
  Gidi: All green octopuses are liars
  Danny: Only a green octopus can be a liar

  
  It is known that only one of these four is a liar, and the rest are truthful.
  a. Who is the liar among the four friends? Explain!
  b. Is it possible to know what his color is?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems
  Additional sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 6

• Question 5 - Quiz

  In a class of 25 students, a quiz was given consisting of 7 questions. Prove that at least one of the following two statements is true:

  1. There is a student who solved an odd number of questions.
  2. There is a question that was solved by an even number of students.
  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question 6 - Pumbaa and the Candies

  Pumbaa has 11 chocolate candies and 13 toffee candies. Each time he can eat either two candies of different types,
  or three candies of the same type. What is the largest number of candies that Pumbaa can eat according to these rules?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 7 - Calculations on the Multiplication Table

  Danny performed the following calculations on the multiplication table of numbers from 1 to 10:
  He calculated the sum of all the numbers in the upper left square of size `9 times 9` within the multiplication table,
  Added to it the sum of all the numbers in the lower right square `9 times 9`,
  Subtracted from the result the sum of all the numbers in the upper right square `9 times 9`,
  And then subtracted the sum of all the numbers in the lower left square `9 times 9`.
  What result did he get?

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  Topics:

  Arithmetic