1972 AMC 12 Problem 17

Attempt Problem 17 of the 1972 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1972 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

17.

A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least

x

times as large as the shorter piece is:

\dfrac12

\dfrac2x

\dfrac1{x+1}

\dfrac1x

\dfrac2{x+1}

Normalize the string length to

1

and let the cut be

t

units from one end

Near either endpoint, solve

1-t\ge xt

Take the string to have length

1.

Near one endpoint, the condition is

1-t\ge xt,

t\le\frac{1}{x+1}.

The same interval occurs at the other endpoint. Their total length, hence the probability, is

\frac2{x+1}.

Therefore, the correct answer is E.

Problem 17 in other years