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Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Advanced Topics #1: Coloring a Tetrahedron

Source:

How many different ways are there to paint the sides of a tetrahedron with exactly

4

colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.

geometry3D geometrytetrahedronrotation

2000 Guts #1: Product of 3 real numbers

Source:

3

real numbers is known to be zero. If the sum of their cubes is

\pi^e

, what is their product equal to?

geometry3D geometry

2000 Algebra #1: Absolute Value Inequalities

Source:

How many integers

x

|x|+5<7

|x-3|>2

inequalitiesabsolute value

2000 HMMT RMT Geometry # 1

Source:

How many rectangles are there on an

8 \times 8

checkerboard? https://cdn.artofproblemsolving.com/attachments/9/e/7719117ae393d81a3e926acb567f850cc1efa9.png

combinatoricsgeometry

2000 Oral #1: Diophantine Equation

Source:

Find all integer solutions to

m^2=n^6+1

2000 HMMT RMT Team #1

Source:

You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is

.7

, what is the smallest number you could have been given? As is customary,

5

’s are always rounded up. Give the answer as a decimal.

number theoryalgebra

2000 HMMT RMT General #1

Source:

a = 2b + c

b = 2c + d

2c = d + a -1

d = a - c

b