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

  Which of these two numbers is larger:

  `A=2011*20122012*201320132013 `  or  `B=2013*20112011*201220122012` ?

  Topics:

  Arithmetic Algebra -> Algebraic Techniques
  Sources:
  • problems.ru web site

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations
  Sources:
  • problems.ru web site

• Question

  Solve the equation:

  `(x + 2010)(x + 2011)(x + 2012) = (x + 2011)(x + 2012)(x + 2013) `

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Find the sum of all natural numbers from `1` to `100`.

  Topics:

  Arithmetic Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• Question

  Given real numbers `a`, `b` and `c` distinct from `0` that satisfy: `a+b/c=b+c/a=c+a/b=1`. Prove that `ab+bc+ca=0`.

  Topics:

  Algebra -> Algebraic Techniques Algebra -> 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

  Given a real number `a` such that `a+1/a` is an integer. Prove that `a^n+1/a^n` is also an integer for every natural number `n`.

  Topics:

  Number Theory Combinatorics -> Induction (Mathematical Induction) Algebra -> Algebraic Techniques Proof and Example Algebra -> Sequences

• Question

  By how much is the sum of all even numbers not exceeding `100` greater than the sum of all odd numbers not exceeding `100`?

  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• Question

  Positive numbers a, b, c, d satisfy `a^3 + b^3 +c ^3 + d^3 >= 3` and also `a^5 + b^5 +c ^5 + d^5 <= 5`

  Prove that `a + b +c + d >= 3 / 2`

   

  Show hint

  Log in here to receive the hint
  Topics:

  Algebra -> Algebraic Techniques Algebra -> Inequalities -> Averages / Means
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 1

• How many solutions does the equation have?

  Given the equation:
  `x^2+2xy+y^2-200x-200y+1900=0`
  How many solutions (x,y) are there, where x and y
  are integers from 1 to 100 (inclusive)?

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations -> Diophantine Equations
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 5