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