Question

Give a proof for each statement.

Answer 1

Answer:

• If a group of 9 kids have won a total of 100 trophies, then at least one of the 9 kids has won at least 12 trophies.
• If a person buys at least 400 cups of coffee in a year, then there is at least one day in which the person has bought at least two cups of coffee.
• The average of three real numbers is greater than or equal to at least one of the numbers.

Explanation:

1)

Suppose that  each kid has less than 12 trophies

Total trophies = 100

Maximum trophies won by one kid = 11

total kids = 9

total number of trophies = 9 * 11 = 99 which contradicts the fact the total number of trophies are 100

2)

Suppose that  person has less than 2 cups of coffee a day

Total cups of coffee = 400

he has bought at least one cup of coffee each day

which means

total number of cups of coffee = 1* 366 = 366 which contradicts the fact the person buys at least 400 cups of coffee in a year

3)

Average of three number = (a+ b+ c)/3

suppose that there are real numbers a, b, and c such  that all three numbers are less than the average of the three numbers.

Let m be the average (a+b+c )/3 = m. Then our assumption states that (a < m) and  (b < m) and (c < m). By adding all the inequalities we get a + b + c < 3m. But m is  defined to be (a+b+c) /3  , so a + b + c = 3m.  But now we have that 3m = a + b + c < 3m. So 3m < 3m which is an obvious  contradiction. Thus our claim is true