Problem set 1 2. Otherwise, at most two people shook hands with person X. Thus, there exist three people who didn' t shake hands with X. Again, there are two further possibilities (a)Among these three, every pair shook hands. Then these three form a triangle (b) Among these three, some pair didn' t shake hands. Then these two and X for a triangle roblem 6. Let a and y be nonnegative real numbers. The arithmetic mean of a and y is defined to be(ar +y)/ 2, and the geometric mean is defined to be vay. Prove that the arithmetic mean is equal to the geometric mean if and only if r Solution Proof. We construct a chain of if-and-only-if implications. The arithmetic mean is equal to the geometric mean if and only if r+y=√ x+y=2√y (x+y)2 -+2 cy+y=4.ry x2-2xy+y2=0 (x-y)2=0 y Problem 7. Use case analysis to prove that all integral solutions to the equation subject to these constraints are in this table 3412 3530 4312 These equations reveal something fundamental about the geometry of our three-dimensional world: we'll revisit them in about three weeks4 Problem Set 1 2. Otherwise, at most two people shook hands with person X. Thus, there exist three people who didn’t shake hands with X. Again, there are two further possibilities: (a) Among these three, every pair shook hands. Then these three form a triangle. (b) Among these three, some pair didn’t shake hands. Then these two and X for a triangle. Problem 6. Let x and y be nonnegative real numbers. The arithmetic mean of x and y is defined to be (x + y)/2, and the geometric mean is defined to be √xy. Prove that the arithmetic mean is equal to the geometric mean if and only if x = y. Solution. Proof. We construct a chain of if­and­only­if implications. The arithmetic mean is equal to the geometric mean if and only if: x + y = √xy x + y = 2√xy 2 ⇔ (x + y) 2 ⇔ = 4xy 2 x 2 ⇔ + 2xy + y = 4xy ⇔ 2 x − 2xy + y 2 = 0 ⇔ (x − y) 2 = 0 ⇔ x − y = 0 ⇔ x = y Problem 7. Use case analysis to prove that all integral solutions to the equation 1 1 1 1 + = + m n e 2 subject to these constraints m ≥ 3 n ≥ 3 e > 0 are in this table: m n e 3 3 6 3 4 12 3 5 30 4 3 12 5 3 30 These equations reveal something fundamental about the geometry of ourthree­dimensional world; we’ll revisit them in about three weeks