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.
Averages / Means-
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脳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?
-
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`
Sources: -
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) `
Sources:Topics:Algebra -> Inequalities -> Averages / Means -
Average of Averages
Consider the following diagram:
Each number in the diagram, connected to 4 other numbers, must be equal to their average:
Sources:
What is the number in the circle marked with a question mark?
Note: The average of four numbers is their sum divided by 4. -
Average of Averages 2
Consider the following diagram:
Each number in the diagram, connected to 4 other numbers, must be equal to their average:
Sources:
What is the number in the circle marked with a question mark? -
Average of Averages 3
Consider the following diagram:
Each number in the diagram, connected to 4 other numbers, must be equal to their average:
Sources:
What is the number in the circle marked with a question mark? -
Shuki's Walk
Shuki walked for 3.5 hours. In every one-hour period, he walked 5 km. Does this mean that Shuki's average speed during the time he walked is 5 km/h?
Sources: -
Question
The numbers from 1 to `10^9` (inclusive) are written on the board. The numbers divisible by 3 are written in red, and the rest of the numbers are written in blue. The sum of all the red numbers is equal to `X`, and the sum of all the blue numbers is equal to `Y`. Which number is larger, `2X` or `Y`, and by how much?
Sources:Topics:Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Number Theory -> Division- Beno Arbel Olympiad, 2017, Grade 8 Question 2
-
THE GROCER AND DRAPER
A country "grocer and draper" had two rival assistants, who prided themselves on their rapidity in serving customers. The young man on the grocery side could weigh up two one-pound parcels of sugar per minute, while the drapery assistant could cut three one-yard lengths of cloth in the same time. Their employer, one slack day, set them a race, giving the grocer a barrel of sugar and telling him to weigh up forty-eight one-pound parcels of sugar While the draper divided a roll of forty-eight yards of cloth into yard pieces. The two men were interrupted together by customers for nine minutes, but the draper was disturbed seventeen times as long as the grocer. What was the result of the race?Sources:Topics:Arithmetic Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means Units of Measurement- Amusements in Mathematics, Henry Ernest Dudeney Question 34