Algebra

Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.

• 6 on the Board

  The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
  Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no more

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 5

• Sum of Squares

  Given a rectangle with an area of 13 and a perimeter of 20. On two adjacent sides of the rectangle, two squares are constructed, as shown in the figure. Find the sum of the areas of the squares.

  

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  Topics:

  Geometry -> Area Calculation Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 7

• Consecutive Numbers

  a. Avi wants to find 10 consecutive numbers whose sum is divisible by 90. Will he succeed?
  b. Benny wants to find 11 consecutive numbers such that their sum is divisible by 90. Will he succeed?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 4

• Numbers on the Board

  The following numbers are written on the board:`1/3,1/2,1,2,3`. In each step, you are allowed to choose any two numbers written on the board and replace each of them with their product.
  Can you reach a quintet of numbers in this way whose sum is `4 1/4`?

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 6

• Sun Minus Five

  In the following arithmetic problems, different letters replace different digits, and identical letters replace
  identical digits.
  `5 = ש + מ + ח`
  `9 = ש + מ + ש`
  Calculate the value of the expression: `ח + ש`

  Topics:

  Arithmetic Algebra -> Equations Puzzles and Rebuses
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 2

• Zionism Street

  On Zionism Street live Benjamin, Chaim, Judah, and Max. Benjamin left his house, walked 169 meters, and arrived at Chaim's house.
  After a short rest, he walked 196 meters and arrived at Judah's house. Then he walked 171 meters and arrived at Max's house.
  Finally, after walking another 144 meters, he returned to his house. What is the distance from Benjamin's house to Judah's house?

  Topics:

  Arithmetic Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 5

• 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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 2

• The Bakery

  In the morning, a bakery's storage room contained 135 kilograms of flour and 92 kilograms of sugar.
  To bake one cake, the baker uses one kilogram of flour and one kilogram of sugar.
  At the end of the workday, the amount of flour remaining for the baker was twice as large as the amount of sugar remaining.
  How many cakes did the baker make during the workday?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 5
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 2

• SLV LVS BLS

  In the following expression, different letters represent different digits, and identical letters represent identical digits:

  SLV = LVS + BLS

  Find the number SLV.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 3

• Sums of Factorials

  For every positive integer n, let us denote `n! = 1*2*3*...*n`

  Find all integers n such that the sum `1! + 2! + 3! + ... + n!` is the square of an integer.

  (Solution format: a,b,c,... i.e., the three smallest numbers separated by commas with no spaces, and three dots after if there are more solutions)

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Algebra -> Sequences
  Sources:
  • Grossman Math Olympiad, 2022, Seniors Question 1