If the whatsis is so when the whosis is is and the so and so is is × so, what is the whosis × whatsis when the whosis is so, the so and so is so × so, and the is is two (whatsis, whosis, is and so are variables taking positive values)?

1975 AMC 12 Problem 14

Attempt Problem 14 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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14.

If the whatsis is so when the whosis is is and the so and so is is

\cdot

so, what is the whosis

\cdot

whatsis when the whosis is so, the so and so is so

\cdot

so, and the is is two (whatsis, whosis, is and so are variables taking positive values)?

whosis

\cdot

\cdot

whosis

so and so

Replace whatsis, whosis, is, and so by

W,H,I,S

The condition

S+S=IS

and positivity force

I=2

The first condition says that

H=I

and

2S=IS

imply

W=S.

Since

S\gt0,

the latter equation gives

I=2.

In the requested case,

H=S,\ 2S=S^2,

and

I=2,

so positivity gives

H=I=S=2.

Hence

W=S=2

and

HW=4=S+S.

Therefore, the correct answer is E.