Algebra, Inequalities

Inequalities are statements comparing two expressions using symbols like <, >, \le, \ge. This topic involves solving inequalities (linear, quadratic, absolute value), proving algebraic or geometric inequalities (e.g., AM-GM), and understanding their properties.

• Question

  Find all natural numbers with the following property: when divided by 7, their remainder is equal to their quotient.

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Arithmetic -> Division with Remainder

• Question

  Find all two-digit numbers `A` such that the square of the sum of its digits is equal to the sum of the digits of `A^2`.

  Topics:

  Number Theory Arithmetic Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Can the product of two consecutive natural numbers be equal to the product of two consecutive even numbers?

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  `10` identical books cost more than `11` dollars, and `9` books of the same type cost less than `10` dollars. How much does one book cost?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Given three real numbers `a`, `b`, and `c`. It is known that `a+b+c>0`, `ab+bc+ca>0`, and `abc>0`. Prove that `a>0`, `b>0`, and `c>0`.

  Topics:

  Algebra -> Inequalities Proof and Example -> Proof by Contradiction

• 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

• Cherries and Blueberries

  `175` kg of cherries cost more than `125` kg of blueberries, but less than `126` kg of blueberries. In addition, it is known that a kilogram of cherries costs a whole number of shekels, and a kilogram of blueberries also costs a whole number of shekels.

  Prove that `80` shekels is not enough to buy one kilogram of blueberries and three kilograms of cherries.

  S. Fomin

  Topics:

  Number Theory Arithmetic Algebra -> Inequalities Algebra -> Word Problems
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 1 Points 3

• Question

  Avi invited guests. They know he lives in apartment `333`, in entrance number `10`, but they don't know which floor. There are `9` floors in the building, and it is known that there are an equal number of apartments on each floor. The guests want to use the elevator. Can they calculate in advance which floor they need to press in the elevator?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Arithmetic -> Division with Remainder

• Question

  The numbers `1`, `2`, `3`, ..., `9` are divided into `3` sets. Prove that there is a set where the product of the numbers is greater than or equal to `72`.

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  Topics:

  Number Theory Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Proof and Example -> Proof by Contradiction

• 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