7th Putnam 1947

Problem B3

Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2?

Solution

If we use polar coordinates, then r = cosec θ - .01, so the length is ∫π/4arctan(1/3) √( (cosec θ - .01)2 + cosec2θcot2θ) dθ. This is obviously horrendous.

The trick is that if we just remove the .01, then the integral gives the curve length for the line segment C, which is 2. But the presence of the -.01 obviously reduces the integrand at every point of the range, so the integral above must have value less than 2.

Comment. One hopes, of course, that the questioner has made it easy by taking the two endpoints a distance greater than 2 apart, or maybe the sum of the distances from each endpoint to the middle greater than 2 (although that is already getting tiresome to check). But no. Or maybe the difference in x-coordinates + the difference in y-coordinates for the endpoints is less than 2 (because the curve length is surely less). But no. I wasted time on all those!