Algebra, Equations

An equation is a statement that two mathematical expressions are equal. Solving an equation involves finding the values of variables that make the statement true. Questions cover various types: linear, quadratic, polynomial, rational, radical, and systems of equations.

• Question

  Given a real number `a` such that `a+1/a` is an integer. Prove that `a^2+1/a^2` is also an integer.

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations Proof and Example

• Question

  Simplify the expression:

  `1/(1-a)+1/(1+a)+2/(1+a^2)+4/(1+a^4)+8/(1+a^8)+16/(1+a^16)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Equations Algebra -> Algebraic Techniques -> Telescoping Sums

• Product of Digits

  For a natural number `x`, let `P(x)` denote the product of the digits of `x`, and let `S(x)` denote the sum of the digits of `x`. 

  How many solutions are there to the equation:

  `P(P(x))+P(S(x))+S(P(x))+S(S(x))=1984`

  Topics:

  Number Theory Algebra -> Equations
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 4 Points 8

• Question

  In a room, there are round chairs with `3` legs and square chairs with `4` legs, totaling `8` chairs. How many of them are square chairs, if the total number of legs in the room is `27`?

  Topics:

  Algebra -> Equations Algebra -> Word Problems

• Question

  In the following arithmetic puzzle, different digits have been replaced by different letters, and identical digits – by identical letters. Reconstruct the puzzle:

  `BAOxxBAxxB=2002`

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Inequalities Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• What is the Smiley?

  In the following exercise, one of the digits has been replaced by a smiley face. Find the digit.

  `1☺xx6-☺=70`

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  Topics:

  Arithmetic Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 2

• 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.

  

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  Topics:

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

• 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
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 5
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 1

• School Supplies

  An eraser and two pencils together cost 19 lira, a pencil and two notebooks together cost 23 lira, a notebook and two
  erasers together cost 18 lira. How much do an eraser, a pencil, and a notebook cost together?

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  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 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