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Here is this week’s problem:

The length and width of a rectangle are integer values. What is the area of the smallest such rectangle that can be inscribed in a circle whose radius is also an integer?

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Rectangle of sides 6×4 can inscribe in a circle of radius 5, a integer value.

1. Diameter must be even.

2. Thus both the sides must be either even or odd.

3. Considering few squares of integers under even 4,16,36,64 and odd 1,9,25,49,81 pairs.

4. Got 36 + 64 =100 whose square root is even = 10, thus circles radius =5.

Area = 24 unit2

Kindly suggest any smarter way, if any.

If the rectangle has sides a and b, and radius of circle is r

(a)^2 + (b)^2 = (2r)^2

a little bit of working around for smallest available squares, and we get

6^2 + 8^2 = 10^2

So the sides of the rectangle are 6 and 8, and area is 48.

(On drawing a diagram, the 3,4,5 triad is very easy to stumble onto– 3 is half the side of first side of rectangle, 4 is half of second side of rectangle, and 5 is radius)