Young Mathematician Olympiad, 2017-2018

• Question from sources: Stage A, Grades 5-6(3)

  Grandma Hannah has many flowerpots with flowers in her house. One day, her three grandchildren, Avi, Beni, and Gili, came to visit and tried to guess how many flowerpots she has.
  Avi said: "Grandma has more than 8 flowerpots,"
  Beni said: "More than 10, I think,"
  And then Gili said: "Grandma Hannah has at least 12 flowerpots."
  "Two of you are right, and one of you is wrong," answered the grandmother. So how many flowerpots does she have in the house?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 3

• Question from sources: Stage A, Grades 5-6(4) - What's the Number?

  Find a three-digit number that, when we add 1 to it, is divisible by 7; when we add 2 to it, is divisible by 8; and when we add 3 to it, is divisible by 9.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 4

• Question from sources: Stage A, Grades 5-6(5)

  Consider all the numbers from 1 to 4242. Find the difference between the number of odd numbers divisible by 3
  and the number of numbers divisible by 7 in this range.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 5

• Question from sources: Stage A, Grades 5-6(6) - Hexagon

  In the diagram, there is a regular hexagon. By what factor is the area of the white region larger than the area of the shaded region?

  (A regular hexagon is a hexagon where all sides are equal and all angles are equal.)

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 6

• Question from sources: Stage A, Grades 5-6(7) - Long Division

  Calculate the value of the expression and write it as a decimal fraction:

  `1/(1+1/(1+1/(1+1/(1+1/(1))))`

  Topics:

  Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 7

• Question from sources: Stage A, Grades 3-4(8), Stage A, Grades 5-6(8) - Differences in the Multiplication Table

  Color the `10xx10` multiplication table with a black and white chessboard coloring, such that the cell of `1xx1` is colored black.
  Find the difference between the sum of all the numbers in the black cells and the sum of all the numbers in the white cells.
   

  	1	2	3	4	5	6	7	8	9	10
  1	1	2	3	4	5	6	7	8	9	10
  2	2	4	6	8	10	12	14	16	18	20
  3	3	6	9	12	15	18	21	24	27	30
  4	4	8	12	16	20	24	28	32	36	40
  5	5	10	15	20	25	30	35	40	45	50
  6	6	12	18	24	30	36	42	48	54	60
  7	7	14	21	28	35	42	49	56	63	70
  8	8	16	24	32	40	48	56	64	72	80
  9	9	18	27	36	45	54	63	72	81	90
  10	10	20	30	40	50	60	70	80	90	100

  Show hint

  Log in here to receive the hint
  Topics:

  Arithmetic Combinatorics -> Number Tables Combinatorics -> Colorings -> Chessboard Coloring
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 8
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 8

• Question from sources: Final, Grades 3-4(1) - Toys

  Jonathan has a collection of wooden toys. Some are cubes and some are spheres, some are red and some are blue.

  It is known that there are more spheres than cubes, and it is known that there are more blue toys than red toys.

  Prove that Jonathan has a blue sphere.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 1

• Question from sources: Final, Grades 3-4(2) - The Grasshopper

  Consider an infinite grid of squares. A grasshopper sits on one of the squares. The grasshopper can jump two squares in any horizontal or vertical direction, and it can jump to the adjacent square diagonally. Can the grasshopper ever reach a square that is adjacent to its starting square by a side?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Combinatorics -> Colorings -> Chessboard Coloring
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 2

• Question from sources: Final, Grades 3-4(3) - Digits in a circle

  The digits from 1 to 5 are written in a circle in some order. Danny summed five two-digit numbers formed from pairs
  of adjacent digits on the circle (clockwise). Find all the possible values for this sum.

  Topics:

  Arithmetic
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 3

• Question from sources: Final, Grades 3-4(4) - Products of Areas

  In the figure, there is a rectangle and a point inside it. Two segments are drawn through the point, parallel to the sides of the rectangle, dividing the rectangle into 4 smaller rectangles.
  Prove that the product of the areas of the shaded rectangles inside the rectangle is equal to the product of the areas of the unshaded rectangles inside the rectangle.

  

  Show hint

  Log in here to receive the hint
  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 4