
The
Problems:

The
Solvers:

Question 20011
submitted by Jaemin Bae, AAST
......

......
1) Eve Drucker, AAST
2) Anna Pierrehumbert, ETHS
3) Michel D'Sa, Highland Park
4) Mukund Ramachandran, Cupertino
5) Anatoly Preygel, Mont. Blair
6) Michael Zhang, Stevenson
7) Daniel Herriges, Highland Park

Question 20012
submitted by Jaemin Bae, AAST
......

......
1) Anna Pierrehumbert, ETHS
2) Michel D'Sa, Highland Park
3) Eve Drucker, AAST
4) Frank Kelly, Fort Worth
5) Michael Erlewine, Highland Park
6) Steven Sivek, TJHSST
7) Gary Sivek, TJHSST

Question 20013
submitted by Michel D'Sa, Highland Park
......

......
1) Steven Sivek, TJHSST
2) Eve Drucker, AAST
3) Anna Pierrehumbert, Evanston
4) Joel Lewis, Stuyvesant
5) Steve Byrnes, Roxbury Latin
6) Jon Pinyan, AAST
7) Al Dracului, Romania

Question 20014
submitted by Gary Sivek, TJHSST
......

......
1) Joel Lewis, Stuyvesant
2) Michel D'Sa, Highland Park
3) Steve Byrnes, Roxbury Latin
4) Anna Pierrehumbert, Evanston
5) Eve Drucker, AAST
6) Jon Pinyan, AAST
7) Michael Erlewine, Highland Park

Question 20015
submitted by Steve Byrnes, Roxbury Latin
......

......
1) Michel D'Sa, Highland Park

Question 20016
submitted by Steve Byrnes, Roxbury Latin
......

......
1) Eve Drucker, AAST
2) Anna Pierrehumbert, Evanston
3) Michel D'Sa, Highland Park
4) Michael Erlewine, Highland Park
5) Joel Lewis, Stuyvesant
6) Howard Yu, Princeton HS
7) John Mangual, Stuyvesant

Question 20017
submitted by Eve Drucker, AAST

......
1) Howard Yu, Princeton HS
2) Michel D'Sa, Highland Park
3) Jaemin Bae, AAST
4) Steve Byrnes, Roxbury Latin

Question 20018
submitted by Suleyman Cengiz
......

......
1) No solutions submitted

Question 20019
submitted by Howard Yu
......

......
1) Michel D'Sa, Highland Park
2) Jon Chu, AAST
3) Steve Byrnes, Roxbury Latin

ANSWERS TO 2001
PROBLEMS
Question 20011
 Use the law of sines to determine that cos^{2}C =
7/8, work from there with law of cosines to obtain
BC^{2}=126.
Question 20012  Let
f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,
and argue that we must have
f(x)=(x1)(x2)(x3)(x4)+2x.
Therefore f(1)=118, the desired quantity.
Question 20013  By
reflecting the figure over successive sides to straighten
the path, we find tan(a)=2Sqrt[3]
and total path length is Sqrt[13].
Question 20014  Using
a 152025 right triangle we can make the perimeter as small
as 80.
Question 20015  One
can prove that the largest distance is no more than 72. Can
you find a configuration of tunnels and caverns that attains
this maximum?
Question 20016 
Routine solid geometry formulas along with the Pythagorean
theorem produce AB=1980.
Question 20017  This
one was tricky. Argue that area(ABC)=15(AB+AC)/2,
then use Ptolemy on ABPC and show that BP=PC=5(BC)/8. This
leads to area(ABC)=480.
Question 20018  One
possible approach is to replace cos^{2}x by
(1+cos(2x))/2, and similarly for the other terms,
then use a summation formula to reduce the equation to (sin
4x)(cos 5x) / (sin x) = 0, yielding the
solutions x=18, 45, 54, or 90 degrees.
Question 20019 
Substitute X=(sin a)(sin b), Y=(sin a)(cos b), and Z=cos b
to reduce to (X+Y2Z)^{2}>=0.
......
