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How many triples of positive integers \( (a, b, c) \) are there such that \[\begin{array} &1\le \frac{a^2 + b^2}{c}\le 3, &a\le 5, &b\le 5, &c\le 5? \end{array}\]

(A) \(\ \ 12\)

(B) \(\ \ 17\)

(C) \(\ \ 19\)

(D) \(\ \ 21\)

(E) \(\ \ 34\)

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\(\text{Pick-}3\) is a lottery in which three numbers are drawn from a larger pool of numbers \(1\) through \(x,\) with the order in which they are drawn irrelevant. If you want the odds of hitting the jackpot to be between \(\frac{1}{2000}\) and \(\frac{1}{3000},\) which of the following numbers is eligible for \(x?\)

\(\begin{array}{r r l}
& \text{I.} & 24\\

& \text{II.} & 26\\

& \text{III.} & 28\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) II only

(C)\(\ \ \) III only

(D)\(\ \ \) I and II only

(E)\(\ \ \) II and III only

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If he does not trust me, he will not see me again.

If he does not see me again, I cannot work for him any more.

If I cannot work for him any more, I will have to look for another job.

Given the above \(3\) statements, which of the following statements must be true?

\(\begin{array}{r r l}
& \text{I.} & \text{If I can keep working for him, then he trusts me.}\\

& \text{II.} & \text{If he sees me again, I do not have to look for another job.}\\

& \text{III.} & \text{If I have to look for another job, then he does not trust me.}\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) I and II only

(C)\(\ \ \) I and III only

(D)\(\ \ \) II and III only

(E)\(\ \ \) I, II and III

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John leaves Chicago bound for New York at \(5\) a.m. He drives at an average speed of \(60\) miles per hour and arrives at a rest area to take a breakfast for half an hour. He leaves the rest area at \(8\) a.m. and drives \(200\) miles before taking one-hour break at another rest area. At \(11\) a.m. John leaves the second area and drives the remaining \(450\) miles straight on, taking only a \(10\)-minute break at the third rest area, until he arrives at New York at \(6\) p.m. What is John's average driving speed between Chicago and the second rest area?

(A) \(\ \ 54 \text{ mph}\)

(B) \(\ \ 60 \text{ mph}\)

(C) \(\ \ 64 \text{ mph}\)

(D) \(\ \ 70 \text{ mph}\)

(E) \(\ \ 80 \text{ mph}\)

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In the figure above, circle \(O\) is tangent to both the \(x\) and \(y\) axes. If \(P=(-1, -2)\) is a point on the circle, which of the following points is NOT on circle \(O?\)

\(\begin{array}{r r l}
& \text{I.} & (-1.5, -8.5)\\

& \text{II.} & (-5, 0)\\

& \text{III.} & (-8, -9)\\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) II only

(C)\(\ \ \) III only

(D)\(\ \ \) II and III only

(E)\(\ \ \) I, II, and III

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