Algebra
Counting
Geometry
Number Theory
Teatime
100
What is (2+4+6)/(1+3+5)-(1+3+5)/(2+4+6)?
7/12
100
Two real numbers are selected independently and at random from the interval [-20,10]. What is the probability that the product of those numbers is greater than zero?
5/9
100
Two tangents to a circle are drawn from a point A. The points of contact B and C divide the circle into arcs with lengths in the ratio 2:3. What is the degree measure of ∠BAC?
36
100
In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b?
224
100
What is 1+1? If you answer correctly, you will also spin the wheel of chaos.
200
Let x and y be two-digit positive integers with mean 60. What is the maximum value of the ratio x/y?
33/7
200
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
729
200
Rectangle ABCD has AB=6 and BC=3. Point M is chosen on side AB so that ∠AMD=∠CMD. What is the degree measure of ∠AMD?
75
200
Let N be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of N?
3
200
Find a function f(x) such that f(x) is irrational for all x∈ℚ.
Example: f(x)=√2
300
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N?
7
300
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
380 or 382 were accepted
300
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
(√2-1)/2
300
In the equation below, A and B are consecutive positive integers, and A, B, and A+B represent number bases: 132A+43B=69A+B. What is A+B?
13
300
Prove Power of a Point for the case where the point is inside the circle (Given chords AB and CD that intersect at point P, show that AP ⋅ BP = CP ⋅ DP)
Join 
and 
.
In 
(Angles subtended by the same segment are equal)
(Vertically opposite angles)
(Corresponding sides of similar triangles are in the same ratio)
400
Two parabolas have equations y=x2+ax+b and y=x2+cx+d, where a, b, c, and d are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?
31/36
400
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?
132
400
A segment through the focus F of a parabola with vertex V is perpendicular to FV and intersects the parabola in points A and B. What is cos(∠AVB)?
-3/5
400
Let n be the smallest positive integer such that n is divisible by 20, n2 is a perfect cube, and n3 is a perfect square. What is the number of digits of n?
7
400
Doomed Yuri: Choose two teams to be linked together (If one team loses 200 points, both teams lose 200 points. If one team has their points doubled, both teams have their points doubled)
500
Let f(x)=1010x, g(x)=log10(x/10), h1=g(f(x)), and hn(x)=h1(hn-1(x)) for integers n≥2. What is the sum of the digits of h2011(1)?
16089
500
A bug travels in the coordinate plane, moving only along the lines that are parallel to the x-axis or y-axis. Let A=(-3,2) and B=(3,-2). Consider all possible paths of the bug from A to B of length at most 20. How many points with integer coordinates lie on at least one of these paths?
195
500
Rhombus ABCD has side length 2 and ∠B=120o. Region R consists of all points inside of the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R?
2√3/3
500
How many positive two-digit integers are factors of 224-1?
12
500
Prove that for ∂u/∂t + (u ⋅ ∇)u = -∇p + vΔu + f(x,t), there exists a smooth, physically admissible force f(x,t) and a smooth initial condition u0(x) such that no smooth solution (u, p) exists for all t≥0, where ∇ ⋅ u = 0 and v>0.