If r>0, then for all p and q such that pq≠0 and pr> qr, we have:

1970 AMC 12 Problem 17

Attempt Problem 17 of the 1970 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 1970 AMC 12 solutions, or check the answer key.

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17.

r\gt0,

then for all

p

and

q

such that

pq\ne0

and

pr\gt qr,

we have:

-p\gt-q

-p\gt q

1\gt-\frac{q}{p}

1\lt \frac{q}{p}

None of these

Because

r\gt0,

the hypothesis gives only

p\gt q

Test the remaining claims with both positive and negative choices of

p,q

Since

r\gt0,

the hypothesis is equivalent to

p\gt q.

Then

-p\lt-q,

so A is false. Taking

(p,q)=(2,1)

makes B and D false. Taking

(p,q)=(1,-2)

makes C false because

-\frac{q}{p}=2.

Thus none of A-D follows in every case.

Therefore, the correct answer is E.