Algebra, Algebraic Techniques

This focuses on the specific methods and skills used to manipulate and simplify algebraic expressions, such as factoring, expanding, collecting like terms, working with fractions, and simplifying radicals. Questions require proficient application of these techniques.

• Question

  The distinct real numbers `x` and `y` satisfy the equation:

  `x^2 – 2000x = y^2 – 2000y`

  Find the sum of `x` and `y`.

  Topics:

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

• Question

  Calculate the product:

  `(1-1/4)(1-1/9)(1-1/16)*...*(1-1/225)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  It is known that `a+b-c=6`, `ac+bc-ab=2`. Find `a^2+b^2+c^2`.

  Topics:

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

• Question

  Calculate the value of the expression (without a calculator):

  `(2^3-1)/(2^3+1)*(3^3-1)/(3^3+1)*(4^3-1)/(4^3+1)*...*(100^3-1)/(100^3+1)`

  Topics:

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

• 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

• Question

  Calculate the sum:

  `1/(1*2)+1/(2*3)+1/(3*4)+...+1/(99*100)`

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums