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# Reasoning - Problem Solving

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) $$\ \ 34$$

$$\text{Pick-}3$$ 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

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

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) $$\ \ 80 \text{ mph}$$

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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