Equations
Geometry
Properties of Numbers
Probability
Miscellaneous
100
Define x@y to be |x-y| for all real numbers x and y. What is the value of (1@(2@3))-((1@2)@3)?
-2
100
What is the area of the shaded figure shown below?
6
100
The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the second number. What is the absolute value of the difference between the first and second numbers?
5
100
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
7/8
100
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or n pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of n?
21
200
What is the sum of all real numbers x for which
|x2-12x+34|=2?
18
200
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?
12-4sqrt(3)
200
The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What is the sum of the digits of n?
6
200
A fair coin is tossed 4 times. What is the probability of getting at least 2 tails?
11/16
200
Mr. Lopez has a choice of two routes to get to work. Route A is 6 miles long, and his average speed along this route is 30 miles per hour. Route B is 5 miles long, and his average speed along this route is 40 miles per hour, except for a 1/2-mile stretch in a school zone where his average speed is 20 miles per hour. By how many minutes is Route B quicker than Route A?
3 3/4
300
There is a positive integer n such that (n+1)!+(n+2)!=n!*440. What is the sum of the digits of n?
10
300
A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
2-sqrt(3)
300
The greatest prime number that is a divisor of 16,384 is 2 because 16,384 = 214. What is the sum of the digits of the greatest prime number that is a divisor of 16383?
10
300
If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?
5/16
300
The taxicab distance between points (x1,y1) and (x2,y2) in the coordinate plane is given by
|x1-x2|+|y1-y2|
For how many points P with integer coordinates is the taxicab distance between P and the origin less than or equal to 20?
841
400
Positive real numbers x and y satisfy y3=x2 and
(y-x)2=4y2. What is x+y?
36
400
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be 4sqrt(2) centimeters, as shown below. What is the area of the original index card?
18
400
The infinite product
evaluates to a real number. What is that number?
sqrt(10)
400
An integer n between 1 and 99, inclusive, is to be chosen at random. What is the probability that n(n+1) will be divisible by 3 ?
2/3
400
At noon on a certain day, Minneapolis is N degrees warmer than St. Louis. At 4:00 the temperature in Minneapolis has fallen by 5 degrees while the temperature in St. Louis has risen by 3 degrees, at which time the temperatures in the two cities differ by 2 degrees. What is the product of all possible values of N?
60
500
For complex number u=a+bi and v=c+di (where i=srqt(-1)), define the binary operation u#v=ac+bdi
Suppose z is a complex number such that z#z=z2+40. What is |z|?
5sqrt(2)
500
A regular hexagon of side length 1 is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these 6 reflected arcs?
3sqrt(3)-pi
500
All the roots of the polynomial 
are positive integers, possibly repeated. What is the value of B?
-88
500
A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
3/7
500
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
-"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.
-"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.
-How many pieces of candy in all did the principal give to the children who always tell the truth?
7