06/05/2006 Challenge Problem

[so19]

06/05/06
Question
If a is nonnegative, is (x^2)+(y^2) > 4a?

(1) (x+y)^2 = 9a
(2) (x-y)^2 = a

The answer for this question says both (1) and (2) individually are insufficient, but (1) and (2) together are sufficient.

I understand (2) is insufficient but I cannot prove that (1) is insufficient. Can someone help me?

Thanks.

[Guest]

If a is nonnegative, is (x^2)+(y^2) > 4a?

(1) (x+y)^2 = 9a

x^2 + y^2 + 2xy = 4a + 5a

if 2xy < 5a then (x^2)+(y^2) > 4a. Not Sufficient

(2) (x-y)^2 = a

x^2 + y^2 - 2xy = 4a - 3a

if 2xy > 3a then (x^2)+(y^2) > 4a. Not Sufficient

Add (1) and (2)

x^2 + y^2 + 2xy = 9a
x^2 + y^2 - 2xy = a
-------------------------------
2 (x^2 + y^2) = 10a

(x^2 + y^2) = 5a > 4a (since a is positve)

[JonathanSchneider]

The anonymous poster above provided an excellent description.
I'd perhaps think about it in simpler terms:

If a is "non-negative," then it could be either 0 or positive. Let's look at these cases separately:

For Statement 1, if a = 0, then the total of x + y must also = 0. However, this does NOT answer our question, because x and y could each be 0 (thus an answer of NO), or x could be negative and y positive, or vice versa (yielding an answer of YES). Furthermore, if we take "a" to have a positive value, say 9, then we can see that x+y must have an absolute value of 9 as well. This will yield an answer of YES to our question above.

[guoliang23]

I think there is some problem with this question:

counter-example "x = y = a = 0" leads to an answer of NO. So even (C) is INSUFFICIENT to answer YES.

Even if we modify the question such that "If a is positive", the correct will NOT be (C). Instead, (A) will be the correct answer:

"x^2 + y^2 + 2xy = 9a", and we know that "x^2 + y^2 >= 2xy", since "x^2 + y^2 - 2xy = (x - y)^2 >= 0". It can be inferred that "x^2 + y^2 >= 4.5a" and "2xy <= 4.5a". So "x^2 + y^2 >= 4.5a > 4a", thus statement (1) is SUFFICIENT.

[esledge]

I think there is some problem with this question:

counter-example "x = y = a = 0" leads to an answer of NO. So even (C) is INSUFFICIENT to answer YES.

Even if we modify the question such that "If a is positive", the correct will NOT be (C). Instead, (A) will be the correct answer:

"x^2 + y^2 + 2xy = 9a", and we know that "x^2 + y^2 >= 2xy", since "x^2 + y^2 - 2xy = (x - y)^2 >= 0". It can be inferred that "x^2 + y^2 >= 4.5a" and "2xy <= 4.5a". So "x^2 + y^2 >= 4.5a > 4a", thus statement (1) is SUFFICIENT.

You are right; there is a problem with this (the zero exception), and changing the constraint to "a is positive" would result in a correct answer of (A), though it is not at all obvious! I'm going to refer this question to our office for correction or deletion. Thank you!

[esledge]

Follow-up:

We think this is a really tough problem as it is, but even tougher if we change the constraint to a > 0. Even though the common quadratics are fair game on the GMAT, we can't imagine the GMAT requiring anyone to realize without a clear hint on (1) that x^2 + y^2 >= 4.5a. Such a problem would be so tricky that it wouldn't serve its true purpose of sorting test takers and assigning scores to them.

However, to keep our challenge problem archives intact, we've edited the problem to make the answer E, keeping variable a non-negative. While the exception case is a still bit of a "gotcha," at least it is x=y=a=0, not some difficult-to-calculate relationship.

Thanks again!