Probability/Counting
Algebra
Geometry
Number Theory
Other
100
John rolls a fair dice and Jack flips a fair coin. If Jack flips a head, then John 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
Suppose r and s are roots of a monic polynomial f(x) satisfying r + s = 10 and rs = 20. What is the value of f(2)?
4
100
Triangle ABC has AB = BC = 10 and CA = 16. The circle Ω is drawn with diameter BC.
Ω meets AC at points C and D. Find the area of triangle ABD.
24
100
The base-nine representation of the number 
is 
What is the remainder when is divided by 
3
100
A number x is chosen uniformly at random from the interval [0, 1], and a number y is chosen uniformly at random from the interval [0, 2]. What is the probability that y > x + 1?
1/4
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
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
150
200
How many of the first 
numbers in the sequence 
are divisible by 
?
505
200
How many strings of length 7 containing only 0s, 1s, and 2s have no two consecutive 0s?
1224
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, like 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
Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon.
80
300
How many distinct values of satisfy 
where 
denotes the largest integer less than or equal to ?
4
300
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