Give a proof for each statement.

(a) 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.

(b) 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.

(c) The average of three real numbers is greater than or equal to at least

one of the numbers.

(d)

There is no smallest integer.

(e) There is no largest odd integer.

(f)

There is no largest rational negative number.

(Note that there is a well-defined ordering of all negative rational

numbers. For example -1/2 is larger than -5, because -1/2 > -5

.)

For all even integers n, n² is a multiple of 4.

(h) For all integers x and y, x² − 4y ‡ 2.

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You can use the following fact in your proof:

If n² is an even integer, then n is also an even integer.

(1) If the product of two positive real numbers is larger than 400, then at

least one of the two numbers is larger than 20.

(i) If the sum of two positive real numbers is larger than 400, then at least

one of the two numbers is larger than 200.

(k) If n is composite, then n has a factor that is greater than 1 and less

than or equal to √n.