Geometry
Number Theory
Algebra
Combinatorics
Probability
100
Name me as many ways to find the area of a triangle as you can! (Each correct answer gets 100 points).
Answers will vary but can include Herons, Base * Height, Product of inradius and the semiperimeter etc.
100
Let a, b, c, d, and e be distinct integers such that
(6−a)(6−b)(6−c)(6−d)(6−e)=45.What is a+b+c+d+e?
25
100
is equal to:
2
100
Using the letters 
, 
, 
, 
, and 
, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" 
occupies position
115
100
Each of two boxes contains three chips numbered 
, 
, 
. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
5/9
200
In a triangle 
, 
, 
and 
. Points and 
lie on 
and 
, respectively, with 
. Points 
and 
are on 
so that 
and 
are perpendicular to . What is the area of a pentagon 
?
240/3
200
What is the remainder when 2025^(2025) is divided by 6?
3
200
, what is the least value that 
can be?
16
200
How many integers between 100 and 1000 contain the digit 2?
251
200
My soccer group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three goals (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?
3/2
300
A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents (Diagram will be provided on the board). The radius of the third circle is:
4/9
300
If n=2^(e1)⋅3^(e2)⋅5^(e3)⋯p^(ek) is the prime factorization of n, how many positive divisors does n have? (Do not have to fully simplify)
(e1+1)(e2+1)(e3+1).....(ek+1)
300
(2√6)/(√2+√3+√5) equals:
√2+√3−√5.
300
How many 
four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?
80
300
Daniel chooses a real number uniformly at random from the interval [0, 2025]. Independently, William chooses a real number uniformly at random from the interval [0, 2050]. What is the probability that William's number is greater than Daniel's number?
3/4
400
Equilateral 
has side length , and squares 
, 
, 
lie outside the triangle. What is the area of hexagon 
?
3 + sqrt (3)
400
There are 
players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest 
players are given a bye, and the remaining 
players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is divisible by what prime number?
11
400
If a≥b>1, what is the largest possible value of loga(a/b) + logb (b/a)? (In this case, since you can't write log base a, just consider loga as log base a and logb as log base b) BE READY TO EXPLAIN THE ANSWER!
0
400
When Martin walks up stairs, he takes one or two steps at a time. His stepping sequence is not necessarily regular. He could go up one step, the two steps, then two again, then one again, etc. How many ways can Martin walk up a 14-step stairwell?
610
400
The number 
has over 
positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
1/19
500
A rectangular piece of paper whose length is 
times the width has area . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area 
. What is the ratio 
?
2:3
500
Find the number of positive integers that are divisors of at least one of 
435
500
Find all values of that have the property that if 
lies on the hyperbola 
, then so does the point 
.
3
500
In a drawer Sandy has 
pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 
socks in the drawer. On Tuesday Sandy selects of the remaining 
socks at random and on Wednesday two of the remaining 
socks at random. The probability that Wednesday is the first day Sandy selects matching socks is 
, where 
and 
are relatively prime positive integers, Find 
.
341
500
A coin is biased in such a way that on each toss the probability of heads is 
and the probability of tails is 
. The outcomes of the tosses are independent. Even Ashoerae has the choice of playing Game A or Game B. In Game A he tosses the coin three times and wins if all three outcomes are the same. In Game B he tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
The probability of winning Game A is 
greater than the probability of winning Game B.