Everyday Math

Reasoning Skills

Reasoning - Problem Solving


How many triples of positive integers (a,b,c) (a, b, c) are there such that 1a2+b2c3,a5,b5,c5?\begin{array}{c}&1\le \frac{a^2 + b^2}{c}\le 3, &a\le 5, &b\le 5, &c\le 5? \end{array}

(A)   12\ \ 12
(B)   17\ \ 17
(C)   19\ \ 19
(D)   21\ \ 21
(E)   34\ \ 34

Pick-3\text{Pick-}3 is a lottery in which three numbers are drawn from a larger pool of numbers 11 through x,x, with the order in which they are drawn irrelevant. If you want the odds of hitting the jackpot to be between 12000\frac{1}{2000} and 13000,\frac{1}{3000}, which of the following numbers is eligible for x?x?

I.24II.26III.28\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 33 statements, which of the following statements must be true?

I.If I can keep working for him, then he trusts me.II.If he sees me again, I do not have to look for another job.III.If I have to look for another job, then he does not trust me.\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 55 a.m. He drives at an average speed of 6060 miles per hour and arrives at a rest area to take a breakfast for half an hour. He leaves the rest area at 88 a.m. and drives 200200 miles before taking one-hour break at another rest area. At 1111 a.m. John leaves the second area and drives the remaining 450450 miles straight on, taking only a 1010-minute break at the third rest area, until he arrives at New York at 66 p.m. What is John's average speed between Chicago and the second rest area (including the first rest time where is speed is 00 miles per hour in the calculation)?

(A)   54 mph\ \ 54 \text{ mph}
(B)   60 mph\ \ 60 \text{ mph}
(C)   64 mph\ \ 64 \text{ mph}
(D)   70 mph\ \ 70 \text{ mph}
(E)   80 mph\ \ 80 \text{ mph}

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

I.(1.5,8.5)II.(5,0)III.(8,9) \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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