WHOLEsome:)
geometry
Probably fun
number theory
logging out
100
At a gathering of 
people, there are 
people who all know each other and 
people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
245
100
Line 
has equation 
and goes through 
. Line 
has equation 
and meets line at point 
. Line 
has positive slope, goes through point 
, and meets at point 
. The area of 
is 
. What is the slope of ?
3/4
100
When 
fair standard 
-sided dice are thrown, the probability that the sum of the numbers on the top faces is can be written as
where 
is a positive integer. What is ?
84
100
How many positive integers satisfy the following condition:
125
100
The solution of the equation 
can be expressed in the form 
. What is 
?
8/7
200
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
29
200
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 
. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
143
200
Chloe chooses a real number uniformly at random from the interval 
. Independently, Laurent chooses a real number uniformly at random from the interval 
. What is the probability that Laurent's number is greater than Chloe's number?
3/4
200
Let 
and 
be two-digit integers such that is obtained by reversing the digits of . The integers and satisfy 
for some positive integer 
. What is 
?
154
200
If log(xy3 ) = 1 and log(x 2y) = 1, what is log(xy)?
3/5
300
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 
?
8
300
In triangle 
the medians 
and 
to sides 
and 
, respectively, intersect in point 
. 
is the midpoint of side 
, and 
intersects in 
. If the area of triangle 
is , then the area of triangle is:
24n
300
A number is randomly selected from the set 
, and a number is randomly selected from 
. What is the probability that 
has a units digit of 
?
2/5
300
The integer is the smallest positive multiple of 
such that every digit of is either 
or 
. Compute 
.
592
300
Positive real numbers 
and have the property that
and all four terms on the left are positive integers, where 
denotes the base- logarithm. What is 
?
10^164
400
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let 
denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of ?
13
400
Triangle has right angle at , and contains a point for which 
, 
, and 
. Find 
.
33
400
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. One such example is
What is the probability that the value of the expression James wrote down is ?
1001/2^14
400
Find all primes 
such that there exist positive integers 
that satisfy 
.
2,3,7
400
What is the value of
21,000
500
Ten identical crates each of dimensions 
. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let 
be the probability that the stack of crates is exactly 
tall, where and are relatively prime positive integers. Find .
190
500
A circle of radius 
has chords 
of length and 
of length 7. When and are extended through and , respectively, they intersect at , which is outside of the circle. If 
and 
, then 
73
500
An urn contains 
green balls and blue balls. A second urn contains 
green balls and 
blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 
. Find .
144
500
How many positive integers less than 
are there such that the equation 
has a solution for ?
412
500
The value of that satisfies 
can be written as 
, where and are relatively prime positive integers. Find 
.
103