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.

Algebraic Techniques Equations Inequalities Word Problems Sequences

• Sum of Products of Digits

  Let `P(n)` denote the product of the digits of the number `n`. For example, `P(1948) = 1 * 9 * 4 * 8 = 288`.

  A.

  Calculate `P(1) + P(2) + P(3) + ... + P(2016)`

  B.

  Find the maximum value of `{P(n)} /n`, where `2016 <= n <= 5777`.

  (Solution format: "x, y/z" where y/z is an unreduced fraction. For example "10000, 35/100")

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  Topics:

  Arithmetic Algebra -> Word Problems Combinatorics -> Number Tables Number Theory -> Division
  Sources:
  • Gillis Mathematical Olympiad, 2016-2017 Question 2

• A 2022x2022 Board and Inversion Operations

  We have a `2022 times 2022` board with real numbers.

  In each move, we can choose a row or a column and a real number `c`.

  Then, we replace each number in the row or column from `x` to `c - x`.

  Is it possible to get from any board to any other board?

   

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  Topics:

  Combinatorics -> Invariants Algebra Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables

• Volume of a Rectangular Prism

  Given a rectangular prism whose face areas are 6, 10, and 15. Find its volume.

  

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 4

• Question

  Find the smallest two-digit number `☺` that satisfies:

  `☺times☺ - ☺ = 600`

  Topics:

  Number Theory Arithmetic Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 5

• Question

  Vered subtracted a number composed of the same digits written in reverse order from a three-digit number.
  As a result, she obtained a two-digit number. Find the two-digit number she obtained.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra Logic -> Reasoning / Logic
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 2

• The Secret Area

  Daniel drew four rectangles whose sides are parallel to each other. The rectangles created four intersection areas (see drawing).
  Given the areas of three of the four. Find the area of the fourth intersection.

  

  Topics:

  Geometry -> Area Calculation Algebra
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 3

• 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
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 6

• Parentheses

  Add parentheses to make the result as large as possible:

  `10000-1000-100-10-1`

  Topics:

  Arithmetic Algebra -> Equations Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 1

• Doughnuts

  Ayala, Benny, Gili, Danny, and Hadas received a package of doughnuts containing

  • 10 doughnuts with dulce de leche
  • 8 doughnuts with peanut butter
  • 9 doughnuts with chocolate
  • 11 with strawberry jam

  Each of them has their favorite type of doughnut.

  • Ayala ate 5 doughnuts of her favorite type
  • Benny ate 6 doughnuts of his favorite type
  • Gili ate 7 doughnuts of her favorite type
  • Danny ate 8 doughnuts of his favorite type
  • Hadas ate 9 doughnuts of her favorite type

  After that, they were left with 3 doughnuts of different types. What is each of their favorite type of doughnut?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 5