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Zarrolou
Is xy < 1?

(1) x + y = 1
The greatest value it can assume is the case in which \(x,y=0.5\) and MAX \(xy=\frac{1}{4}<1\).
Sufficient

(2) x^2 + y^2 = 1
Same thing here, the gratest value xy can have is if the terms are equal \((\frac{1}{2})\), we take their positive root (to max the multiplication) so \(x,y=\sqrt{\frac{1}{2}}\); also in this case \(xy\) is less than 1 (\(xy=\frac{1}{2}\) at most).
Sufficient

Hi Bunuel, can you please confirm the rule I've used? I remember it from old books, but I am not 100% sure of it
Thanks in advance

Yes. The property you are referring to is: for given sum of two numbers, their product is maximized when they are equal.

Hope it helps.
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Zarrolou
Is xy < 1?

(1) x + y = 1
The greatest value it can assume is the case in which \(x,y=0.5\) and MAX \(xy=\frac{1}{4}<1\).
Sufficient

(2) x^2 + y^2 = 1
Same thing here, the gratest value xy can have is if the terms are equal \((\frac{1}{2})\), we take their positive root (to max the multiplication) so \(x,y=\sqrt{\frac{1}{2}}\); also in this case \(xy\) is less than 1 (\(xy=\frac{1}{2}\) at most).
Sufficient

Hi Bunuel, can you please confirm the rule I've used? I remember it from old books, but I am not 100% sure of it
Thanks in advance

Yes. The property you are referring to is: for given sum of two numbers, their product is maximized when they are equal.

Hope it helps.

Questions about this concept:
if-x-2-y-2-100-and-x-0-and-y-0-the-maximum-value-of-x-147064.html
if-the-population-of-city-a-is-increased-by-a-in-year-130222.html
the-discreet-charm-of-the-ds-126962-20.html#p1039634
is-x-2-y-108343.html
maxmize-an-ear-of-a-right-triangle-128319.html
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stunn3r
Is xy < 1?

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

Maybe another approach which is a bit visual:

F.S 1:Imagine the straight line on the x-y axis : Omitting the x/y intercepts where anyways the value of xy = 0 ,any point on the line in the 2nd quadrant will always have the sign of (x,y) as (-,+)--> The product xy<0 and hence xy<1. Similarly for any point on the line in the 4th quadrant, where (x,y) will always be (+,-) and xy<1. For any point on the line in the first quadrant except (1,0) or (0,1), 0<x<1 and 0<y<1 --> 0<xy<1. Sufficient.

F.S 2:Similarly, for the circle, the same analogy for different sign of (x,y) would be in place for the 2nd and 4th quadrant. IN the 1st and 3rd quadrant,for the reason given above, the value of xy can never be greater than 1 as they are individually less than 1.Sufficient.
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