Combo
Algebra
Geometry
Number Theory
Other
100

John rolls a fair dice and Jack flips a fair coin. If Jack flips a head, then John's gets money equivalent to triple his dice roll. If Jack flips a tail, then John gets nothing. What is the expected value of John's winnings?

21/4

100

What is the least possible value of  for real numbers  and ?

1

100

In a plane, four circles with radii  and  are tangent to line  at the same point  but they may be on either side of . Region  consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region ?

65pi

100


The base-nine representation of the number  is  What is the remainder when  is divided by

3

100

How many strings of length 7 containing only 0s, 1s, and 2s have no two consecutive 0s?

1244

200

Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

56

200

Consider the set of all fractions , where  and  are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by , the value of the fraction is increased by ?

1

200

All three vertices of  are lying on the parabola defined by , with  at the origin and  parallel to the -axis. The area of the triangle is . What is the length of ?

8

200

How many of the first  numbers in the sequence  are divisible by ?

505

200

Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a  false positive rate--in other words, for such people,  of the time the test will turn out negative, but  of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let  be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to ?

1/11

300

Consider the paths of length  that follow the lines from the lower left corner to the upper right corner on an  grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.

294

300

Let  be the unique polynomial of minimal degree with the following properties:

  •  has a leading coefficient ,
  •  is a root of ,
  •  is a root of ,
  •  is a root of , and
  •  is a root of .

The roots of  are integers, with one exception. The root that is not an integer can be written as , where  and  are relatively prime integers. What is ?

47

300

In triangle , , , and . Distinct points , , and  lie on segments , , and , respectively, such that , , and . The length of segment  can be written as , where  and  are relatively prime positive integers. What is ?

21

300

How many distinct values of  satisfy , where  denotes the largest integer less than or equal to ?

4

300

Let , , and  be positive integers with    such that  and .

What is ?

253

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