[/quote]
,
and 
.
.
of degree 
.![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of 
![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
and 
Prove that
Let 
and 
Prove that
not isosceles, the incircle 
.
is the intersection of 
.
is the projection of 
onto 
.
cut 
.
is the projection of 
.
cut 
respectively. Prove 
are collinear
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
intersects 
at 
.
cuts segment 
at 
respectively. If 
and 
,
equals 
.
are points on a circle 
,
,
.
at 
is on 
such that 
.
intersects 
.
is the reflection of 
.
lies on the same circle.
Y by Adventure10
Juli is a mathematician and devised an algorithm to find a husband. The strategy is:
• Start interviewing a maximum of
prospective husbands. Assign a ranking 
to each person that is a positive integer. No two prospects will have same the rank .
• Reject the first
men and let 
be highest rank of these men.
• After rejecting the first men, select the next prospect with a rank greater than and then stop the search immediately. If no candidate is selected after 
interviews, the 
person is selected.
Juli wants to find the value of for which she has the highest probability of choosing the highest ranking prospect among all candidates without having to interview all prospects.
(a) (6 points:) What is the probability that the highest ranking prospect among all prospects is the 
prospect?
(b) (6 points:) Assume the highest ranking prospect is the person to be interviewed. What is the probability that the highest rank candidate among the first 
candidates is one of the first candidates who were rejected?
(c) (6 points:) What is the probability that the prospect with the highest rank is the
person and that Juli will choose the man using this algorithm?
(d) (16 points:) The total probability that Juli will choose the highest ranking prospect among the prospects is the sum of the probability for each possible value of 
with ranging between 
and .
Find the sum. To simplify your answer use the formula
(e) (6 points:) Find that value of that maximizes the probability of choosing the highest ranking prospect without interviewing all candidates. You may need to know that the maximum of the function 
is approximately 
, where 
is a constant and 
is Euler’s number, 
• Start interviewing a maximum of
prospective husbands. Assign a ranking 
to each person that is a positive integer. No two prospects will have same the rank .• Reject the first
men and let 
be highest rank of these men.• After rejecting the first
men, select the next prospect with a rank greater than and then stop the search immediately. If no candidate is selected after 
interviews, the 
person is selected.Juli wants to find the value of
for which she has the highest probability of choosing the highest ranking prospect among all candidates without having to interview all prospects.(a) (6 points:) What is the probability that the highest ranking prospect among all
prospects is the 
prospect?(b) (6 points:) Assume the highest ranking prospect is the
person to be interviewed. What is the probability that the highest rank candidate among the first 
candidates is one of the first candidates who were rejected?(c) (6 points:) What is the probability that the prospect with the highest rank is the
person and that Juli will choose the man using this algorithm?(d) (16 points:) The total probability that Juli will choose the highest ranking prospect among the
prospects is the sum of the probability for each possible value of 
with ranging between 
and .Find the sum. To simplify your answer use the formula
(e) (6 points:) Find that value of
that maximizes the probability of choosing the highest ranking prospect without interviewing all candidates. You may need to know that the maximum of the function 
is approximately 
, where 
is a constant and 
is Euler’s number, 
This post has been edited 1 time. Last edited by soyeb.jim, Mar 3, 2018, 2:04 PM
