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Part of 2006 Harvard-MIT Mathematics Tournament

Problems(4)

2006 Algebra #1: Swimming Through Cambridge

Source:

Larry can swim from Harvard to MIT (with the current of the Charles River) in

40

minutes, or back (against the current) in

45

minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in

15

minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

HarvardcollegeMIT

2006 Calculus #1: Functional Equation in Derivatives

Source:

A nonzero polynomial

f(x)

with real coefficients has the property that

f(x)=f^\prime(x)f^{\prime\prime}(x)

. What is the leading coefficient of

f(x)

calculusderivativealgebrapolynomial

2006 Combinatorics #1: Vernonia High School

Source:

Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that

74

are on the football team;

26

are on the baseball team;

17

are on both the football and lacrosse teams;

18

are on both the baseball and football teams; and

13

are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.

2006 Geometry #1: Equiangular Octagon

Source:

ABCDEFGH

is equiangular. Given that

AB=1

BC=2

CD=3

DE=4

EF=FG=2

, compute the perimeter of the octagon.

geometryperimeter