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A certain closed curve consists of n semicircles having the same radius of $$\frac{r}{n}$$ together with a larger semicircle having a radius of r. The n semicircles are aligned along the diameter of the larger semicircle as indicated in the figure for n=7. The area of the region enclosed by the curve, in terms of n, and r, is which of the following?


As shown in the figure, the centers of the two circles are respectively on the circumference of each other, and the radius is 3. What is the perimeter of this figure?
The sum of 101 consecutive even integers is 20,200.

Quantity A

The average (arithmetic mean) of the 101 integers

Quantity B

The median of the 101 integers


Quantity A

$$\frac{100!}{99!}$$

Quantity B

$$\frac{ (100!-99!)}{98!}$$


n is an even integer greater than 2

Quantity A

$$\frac{n!}{(\frac{n}{2})!}$$

Quantity B

2($$\frac{n}{2}$$)!




If Mark randomly walks from point P to point R (he can only walk either rightward or upward), what is the probability that he passes through point Q?

Give your answer as a fraction.
Three numbers are to be selected at random and without replacement from the five numbers 4, 5, 7, 8 and 11. What is the probability that the three numbers selected could be the lengths of the sides of a triangle?
Some norms governing authors' dedications of their works are (i)___________ to the norms governing ordinary speech (it may be wrong to lie, and thus wrong to lie in dedication), but other norms (ii)__________ dedications. For example, one ought not dedicate to a worthy recipient an object that one feels is devoid of value.
Although collisions between the largest planetesimal belt are _________, they can release large amounts of debris when they occur.
Although economics emerged as a discipline in Great Britain and France at the end of the eighteenth century, it has taken two centuries to become (i)____________ field of study: among early economists, intuition and opinion (ii)____________ facts and figures in economic thought, but the teaching of economics today is dominated by complex algorithms and mathematical models.
After the first monarch butterfly overwintering site was discovered in Mexico in 1975, the biologist Lincoln Brower (i) _____________ the (ii) _____________ of additional sites. The principal reason for Brower`s contention was that roosting in a very limited area would make the species highly vulnerable to wildfire.
Science is a seemingly inexhaustible source of _____________, yet paradoxically scientists have been portrayed in recent decades as rather conservative--resisting innovation in their discipline sometimes to the point of dogmatism.
The overall message of this book on neuroscience is (i) _____________: the brain is quite (ii) _____________, and therefore damaged brains can be healed, aging brains can be rejuvenated, and even ordinarily healthy brains can be made faster and better.
Which of the following best describes the function of the highlighted sentence in the context of the passage as a whole?
The primary purpose of the passage is to
The author mentions "energy-rich honey dew primarily to
The author mentions the "cohort" primarily to
What is the reminder when $$ (345,606)^{2}$$ is divided by 20?
What is the value of $$\frac{51!-50!}{50!-49!}$$?

Give your answer as a fraction.

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