Algebra, Algebraic Techniques, Short Multiplication Formulas / Algebraic Identities

These are specific algebraic identities like `(a+b)^2 = a^2+2ab+b^2`, `(a-b)^2 = a^2-2ab+b^2`, and `a^2-b^2 = (a-b)(a+b)`. Questions involve recognizing when to apply these formulas for quick expansion, factorization, or simplification of expressions.

• Question

  Simplify the expression and find its value:

  `1987/(198719871987^2-198719871988*198719871986)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities

• 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

• Question

  Which number is larger: `sqrt(2016+2017)` or `sqrt(2016)+sqrt(2017)`?

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Inequalities Algebra -> Algebraic Techniques -> Roots / Radicals

• Question

  Find the value of the expression without a calculator:

  `(sqrt(sqrt(2)-1))/(sqrt(sqrt(2)+1))+ (sqrt(sqrt(3)-sqrt(2)))/(sqrt(sqrt(3)+sqrt(2)))+ (sqrt(2-sqrt(3)))/(sqrt(2+sqrt(3)))`

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  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Algebraic Techniques -> Roots / Radicals Algebra -> Algebraic Techniques -> Multiplication by the Conjugate

• Numbers Squared

  Ayala took all the numbers divisible by 3 from 1 to 99, inclusive, squared each of them, summed the results, and multiplied the resulting number by 2.
  The duck took all the numbers not divisible by 3 from 1 to 100, inclusive, squared each of them, and summed the results.
  What is the difference between the number the duck obtained and the number Ayala obtained?

  Topics:

  Arithmetic Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 4