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

  Calculate the product:

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

  Topics:

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

• Question

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

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

  Topics:

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

• 50 to the Power of

  Show that in the rightmost 504 digits of `1+50+50^2+...+50^1000`

  Each digit appears a number of times divisible by 12

  Show hint

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Algebraic Techniques -> Telescoping Sums
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 2

• Partitioned Roots

  Find the sum `1/{sqrt1+sqrt2}+1/{sqrt2+sqrt3}+...+1/{sqrt99+sqrt100}`.

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  Topics:

  Algebra -> Algebraic Techniques -> Telescoping Sums Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 2

• Question

  Given distinct rational numbers a, b, c, prove that `sqrt{1/(a-b)^2+1/(b-c)^2 +1 /(c-a)^2}`

  is rational.

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  Topics:

  Number Theory Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 3

• Question

  Given natural numbers m, n such that `m/n <= sqrt 23`, prove that `m/n+3/{mn} <= sqrt 23`

  Show hint

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Algebra -> Inequalities Proof and Example -> Proof by Contradiction Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 4

• The Matchstick Root

  Move one matchstick to make a correct equality:

  

  Topics:

  Algebra -> Algebraic Techniques -> Roots / Radicals Puzzles and Rebuses -> Matchstick Puzzles

• Volume of a Rectangular Prism

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

  

  Show hint

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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

• Quadratic Equations

  Solve the equations:

  a: `({x^2+6}/{x^2-4})^2=({5x}/{4-x^2})^2`

  b: `7(x+1/x)-2(x^2+1/x^2)=9`

  c: `sqrt{x+2sqrt{x-1}}-sqrt{x-2sqrt{x-1}}=2`

  Topics:

  Algebra -> Equations Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 1