Algebra
Probability and Counting
Expected Value
Number Theory
100
Without paper, what is 25 x 375
9375
100
What is the probability that Angela gets an 100 as her overall trig grade for the year?
0 percent
100
Whats the expected value of a dice roll?
3.5
100
If 
, what is the value of 
?
11
200
Suppose 
. Find all possible values of 
.
.
200
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
31/432
200
A counter begins at 0. Then, every second, the counter either increases by 1 or resets back to 0 with equal probability. The expected value of the counter after ten seconds can be written as m n , where m, n are positive integers and gcd(m, n) = 1. Find 100m + n.
103324
200
In base 
, the number 
ends in the digit 
. In base 
, on the other hand, the same number is written as 
and ends in the digit 
. For how many positive integers 
does the base--representation of end in the digit ?
13
300
Simplify the expression 
.
1
300
Three points are selected randomly on the circumference of a circle. What is the probability that the triangle formed by these three points contains the center of the circle?
1/4
300
At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form โsectorsโ broken up by the taken slices, e.g. if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed?
11/3
300
Determine all non-negative integral solutions 
if any, apart from permutations, of the Diophantine Equation 
.
No solution
400
If 
, then what is 
?
81/64
400
Roger and Stacy each go to the county fair on the same day. They each separately show up at a random time between 12:00 and 6:00. Roger stays for an hour, whereas Stacy stays for 2 hours. If we know that at some time they were both at the fair simultaneously, what is the probability that they were both there at exactly 3:00?
4/31
400
You are the first lucky player to play in a slightly modified episode of Deal or No Deal! Initially, there are sixteen cases marked 1 through 16. The dollar amounts in the cases are the powers of 2 from 2 1 = 2 to 216 = 65536, in some random order. The game has eight turns. In each turn, you choose a case and claim it, without opening it. Afterwards, a random remaining case is opened and revealed to you, then removed from the game. At the end of the game, all eight of your cases are revealed and you win all of the money inside them. However, the hosts do not realize you have X-ray vision and can see the amount of money inside each case! What is the expected amount of money you will make, given that you play optimally?
1835012/15
400
Let ๐ be the set of all positive integer divisors of 100,000 . How many numbers are the product of two distinct elements of ๐?
117