If and are positive, which of the following must be greater than ? I. II. III.
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This one took me way longer than it should have. I kept trying to plug in x=1, y=1 and that made II and the target equal I think, so I had to pick different numbers. Definitely a trap if you only test equal values.
Same here, x=y=1 makes I and II both suspiciously nice. I switched to x=1, y=4 and things cleared up.
Yeah the equal-values trap got me too. Lesson learned: always test unequal positives on these.
Quick question on III — since x and y are positive, can sqrt(x) - sqrt(y) ever be negative? If so, then it can't "must be greater," right? That's how I eliminated it fast.
Exactly, if y > x the numerator goes negative while the target stays positive, so III is out.
I rationalized I by multiplying numerator and denominator by something to compare it to 1/sqrt(x+y). Bit of algebra but it works. II is the sneaky one because it looks small but the numerator is actually bigger than sqrt(x+y) when you square it.
Nice, squaring both sides for II is way cleaner than what I did. Thanks.
655-705 feels about right, this is more about careful elimination than heavy computation. Got it down to II only after a couple of minutes.
What it tests
Your handling of square and higher roots, including simplifying radicals.
Common trap
Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.