and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
such that 
.
.
has 
,
,
.
.
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
holds inequality 
![\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}}
\, \mathrm{d}u
\]](http://latex.artofproblemsolving.com/9/1/1/91165e6b531233c56b799ed32ded982d57b6c87a.png)
such that
diverges for 
(harmonic sequence) but converges for 
because 
,
converges
![$f(x)\in R[x]$](http://latex.artofproblemsolving.com/a/1/5/a15dc97c051e81ae9ff0c060eaa4367bc723ccc9.png)
show that 
has at least one root of multiplicity one
Y by
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7
A, B, C, D ... all the way up to Z are all different numbers from 1 to 26. What's the largest possible value of (A-x)(B-x)(C-x)...(Z-x)?
2
A "Rizzler Number" is a number such that the sum of all of its odd factors, minus the number of factors, is even. Figure out a method to find any Rizzler Number from the previous one.
3
Bobby draws a polynomial who's side lengths are the powers of 2 from 1 to 64. What is the center of the largest circle that can fit inside it?
4
In Equalville, everyone gets 50 coins a year, and in Baronville, some get 70 while some get 30. However, Equalville is ruled by the Lord Potato, who commits tax fraud and takes 1434 coins instead. Because all the payment is in 50, 70, and 30, there are only 3 coins: one worth 50, one worth 30, and one worth 70. How do you pay the Lord Potato?
5
There are 6 children, Amelia, Bob, Carly, Dave, Ellen, and Fred, who are all sitting in a row. Amelia must sit next to Carly and Dave, Bob must sit next to Ellen, And Dave hates Carly and Fred and wants to be at least 3 spaces away from both of them. How many ways are there to arrange the children? Prove that this grows exponentially as more children are added, with no preferences.
6
Maria fails to make AIME. How does she convince her parents that it was reasonable to study 25 hours a day, instead of 26?
7
April Fools - all of these are troll problems. Enjoy thinking through them.
