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

  On Alice's birthday, her friends are asked how old she is. The Mad Hatter says that Alice's age is greater than `11`, and the Cheshire Cat says that her age is greater than `10`. It is known that exactly one of them is wrong. How old is Alice now? Explain!

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  Topics:

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

• Question

  a. The apple is heavier than the peach, and the peach is heavier than the pear. Which is heavier – the apple or the pear?

  b. The orange is heavier than the pear, and the plum is lighter than the orange. Which is heavier – the pear or the plum?

  Topics:

  Arithmetic Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Question

  Given that the number `A` is positive, the number `B` is negative, and the number `C` is equal to `0`. What is the sign of the number `A*B+A*C+B*C`?

  Topics:

  Arithmetic Algebra -> Inequalities

• Question

  In a square with side length 1, a finite number of segments parallel to the sides of the square were drawn, with a total length of 18 (they can intersect). Prove that among the parts into which the square is divided by the segments, there is a part with an area of at least 0.01. 
  A. Engenes, A. Brazins 

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Inequalities Proof and Example -> Proof by Contradiction
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 6

• Apples and Pears

  There is a basket containing `30` fruits. It is known that among any `12` fruits we take from the basket, there is necessarily at least one apple, and among any `20` fruits there is necessarily one pear. How many apples and how many pears are there in the basket?

  Topics:

  Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Corruption in Parliament

  The parliament of the magical country consists of `20` people. It is known that among the `20` members of parliament, there is at least one who is not corrupt. Additionally, it is known that for any two members of parliament we choose, one is necessarily corrupt. How many members of parliament of the magical country are corrupt?

  Topics:

  Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Shlomi's Schedule

  Shlomi dedicates `1/3` of his time to playing games on the tablet, `1/5` of his time to studying at school, `1/6` to watching movies, `1/70` to solving mathematical problems, and `1/3` to sleeping.

  Is it possible to live like that?

  Topics:

  Algebra -> Inequalities Arithmetic -> Fractions
  Sources:
  • problems.ru web site

• Question

  Which number is larger:

  `2^300` or `3^200`?

  Topics:

  Arithmetic Algebra -> Inequalities

• Question

  A box contains pencils in three colors: red, green, and blue, totaling `20` pencils. There are `6` times more blue pencils than green pencils. There are fewer red pencils than green pencils. How many red pencils are in the box?

  Topics:

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

• Birds and Seeds

  Nine identical birds eat less than `1001` seeds for lunch, and ten such birds eat more than `1100` seeds for lunch. How many seeds does one bird eat for lunch?

  Topics:

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