Algebra, Inequalities, Averages / Means

Averages or means are measures of central tendency for a set of numbers. Common types include arithmetic mean (sum divided by count), geometric mean, and harmonic mean. Questions involve calculating these, solving word problems with averages, and understanding their properties and comparisons.

• Question

  Shuki walked for 3.5 hours. In each hour-long period, he traveled 5 km. Does it follow that Shuki's average speed during this entire time was 5 km per hour?

  Topics:

  Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means

• Question

  Can you find `35` integers whose average is equal to `6.35`?

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  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Inequalities -> Averages / Means Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• Question

  The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site

• Question

  Given an `M times N` matrix, where each cell contains a real number. It is known that the sum of the numbers in each row and each column is equal to `1`.

  Prove that `M = N`.

  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Algebra -> Inequalities -> Averages / Means

• Question

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• 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`

   

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  Topics:

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

• Question

  For all `a,b,c >=1 ` that satisfy `a+b+c= 2abc `

  Prove that `root (3) ((a+b+c)^2) >= sum_{cyc} root (3) (ab-1) `

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  Topics:

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

• Average of Averages

  Consider the following diagram:

  

  Each number in the diagram, connected to 4 other numbers, must be equal to their average:
  What is the number in the circle marked with a question mark?
  Note: The average of four numbers is their sum divided by 4.

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 1

• Average of Averages 2

  Consider the following diagram:

  

  Each number in the diagram, connected to 4 other numbers, must be equal to their average:
  What is the number in the circle marked with a question mark?

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 1

• Average of Averages 3

  Consider the following diagram:

  

  Each number in the diagram, connected to 4 other numbers, must be equal to their average:
  What is the number in the circle marked with a question mark?

  Topics:

  Algebra -> Equations Algebra -> Inequalities -> Averages / Means
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 1