
The
Problems:

The
Solvers:

Question 20021
Submitted by Jaemin Bae, AAST
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1) Pandit Ruse, Stuyvesant
2) Jonathan Gray, Gray Homeschool
3) Eve Drucker, AAST
4) Simon RubinsteinSalzedo, Homestead
5) Joel Lewis, Stuyvesant
6) Howard Tong, Home School
7) Brian Rice, Marion Sr HS

Question 20022
Submitted by Joel Lewis, Stuyvesant
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1) Matthew Welsh, Stuyvesant
2) Howard Yu, Princeton HS
3) Howard Tong, Tong Home School
4) Jeff Nanney, Clark High School
5) Steve Byrnes, Roxbury Latin
6) Jaemin Bae, AAST
7) Yifei Chen, W.W. Plainsboro North

Question 20023
Submitted by Matthew Welsh, Stuyvesant
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......
1) Steve Byrnes, Roxbury Latin
2) Joel Lewis, Stuyvesant
3) Andrew Glazer, Glenbrook North
4) Jaemin Bae, AAST
5) Yifei Chen, W.W. Plainsboro North
6) Darwin Candiver, Davis Aerospace
7) Michael Erlewine, Highland Park Sr HS

Question 20024
Submitted by Steve Byrnes, Roxbury Latin
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......
1) Matthew Welsh, Stuyvesant
2) Brian Rice, Marion Senior HS
3) Darwin Candiver, Davis Aerospace
4) Joel Lewis, Stuyvesant
5) Yifei Chen, W.W. Plainsboro North
6) Jaemin Bae, AAST
7) Jonathan Mizrahi, TJHSST

Question 20025
Submitted by Anna Pierrehumbert, Evanston
Township
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1) Matthew Welsh, Stuyvesant
2) Andrew Laitman, Stuyvesant
3) Jaemin Bae, AAST
4) Brian Rice, Marion Sr HS
5) Joel Lewis, Stuyvesant
6) Chris Breaux, Jesuit HS
7) Yifei Chen, W.W. Plainsboro North

Question 20026
Submitted by Brian Rice, Marion Senior HS
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......
1) Joel Lewis, Stuyvesant
2) Sam Cross, Stuyvesant
3) Darwin Candiver, Davis Aerospace
4) Steve Byrnes, Roxbury Latin

Question 20027
Submitted by Nick Boatman, Carter High
School
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......
1) Matthew Welsh, Stuyvesant
2) Jaemin Bae, AAST
3) Joel Lewis, Stuyvesant
4) Tim Abbott, Thomas Jefferson
5) Atoshi Chowdhury, Baton Rouge
6) Andrew Laitman, Stuyvesant
7) Yifei Chen, W.W. Plainsboro North

Question 20028
Submitted by a teacher from Kazakstan
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......
1) Matthew Welsh, Stuyvesant
2) Jaemin Bae, AAST
3) Nick Boatman, Carter HS
4) Tim Abbott, Thomas Jefferson
5) Joel Lewis, Stuyvesant
6) Yifei Chen, W.W. Plainsboro North
7) Atoshi Chowdhury, Baton Rouge

ANSWERS TO 2002
PROBLEMS
Question 20021
 Use your favorite method (Menelaus, for example) to find
that AP/PA'=25/9. Then compute AA'=12/5 by finding the area
in two different ways to wind up with AP=30/17.
Question 20022  Chase
angles in the cyclic quadrilaterals to argue that the only
possibility is that all the triangles are equilateral. Hence
area(ABC) = 4*Sqrt[3].
Question 20023  The
answer is 2*Sqrt[3]3. To see why, view the
complete solution submitted by
the proposer.
Question 20024  The
question is equivalent to asking in how many ways six
distinguishable objects may be grouped into one or more
piles. There are 203 ways to accomplish this.
Question 20025  Use a
partial fraction decomposition to rewrite (x1)/x(x+1)^2 as
2/(x+1)^2+1/(x+1)1/x. Using the fact that many of the terms
in the resulting series cancel, and that 1/1+1/4+1/9+... =
Pi^2/6, one obtains the final answer of Pi^2/33.
Question 20026  This
solution requires some fairly sophisticated combinatorics.
We will be content to give the answer, which is C(n,
2)*C(2t, t)/(2^(2t)).
Question 20027  The
remainder when P(x) is divided by (xa)^2 is just
(xa)*P'(a)+P(a). (Can you prove it?) Therefore
P(x)=(xa)^2*Q(x)+(xa)*P'(a)+P(a), so the answer is
1776x1549.
Question 20028  The
hint follows from the fact that abc=1 and
(a^3b^3)(a^2b^2)>=0. Similar results hold if we
exchange c with either a or b. Adding these three
inequalities proves the claim.
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