Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6

• Question 1 - Ten-Digit Number

  Yael writes ten-digit numbers in whose decimal representation each of the digits `0, 1, 2, 3, 4, 5, 6, 7, 8, 9` appears exactly once.
  In the numbers that Yael writes, the difference between any two adjacent digits is at least 2. What is the smallest number Yael can write?
   

  Topics:

  Number Theory Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 3

• Question 2 - How Much is Happy?

  In the following arithmetic exercises, different letters replace different digits, and identical letters replace identical digits.
  `19 = ט + מ + ח + ש`
  `10 = ק + ח + ש + מ`
  Calculate the value of the expression: `ח + מ + ש`

  Topics:

  Arithmetic Algebra -> Equations Puzzles and Rebuses

• Question 3 - Orange Star of David

  The area of the blue triangle is equal to 1. Calculate the area of the orange Star of David:

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

• Question 4 - Five Numbers

  Given five distinct positive integers. The sum of these numbers is 27. Additionally, it is known that the product of these five numbers is odd. Calculate this product.

  Topics:

  Arithmetic Number Theory -> Division -> Parity (Even/Odd)

• Question 5 - Angles

  Calculate the sum of the marked angles:

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation

• Question 6 - More Game Cubes

  Aviv has game cubes, where two opposite faces of each are painted red and the rest are blue.
  Aviv glued together a `3 xx 3 xx 3` cube from the game cubes. Then his friend Kfir came and calculated the total red area on the surface of the large cube.
  What is the largest result Kfir can get?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Combinatorics -> Combinatorial Geometry Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings

• Question 7 - More Numbers with Ascending Digits

  Miri writes down all the numbers whose decimal representation contains only the digits `1, 2, 3, 4, 5, 6`.
  (Not all digits must appear) and all the digits that appear are written in ascending order (e.g., 1356 or 124 or 5 but not 162 and not 1223).
  How many numbers will Miri write down?

  Topics:

  Combinatorics -> Product Rule / Rule of Product Set Theory