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Part of 2009 Harvard-MIT Mathematics Tournament

Problems(3)

2009 Algebra #1 - Difference of Squares

Source:

a

b

are positive integers such that

a^2-b^4= 2009

a+b

2009 Calculus #1: One Unit Lower

Source:

f

be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of

f

always meet the

y

-axis 1 unit lower than where they meet the function. If

f(1)=0

f(2)

calculusfunctionanalytic geometrygraphing linesslope

2009 Combinatorics #1 - Arranging Integers from -7 to 7

Source:

How many ways can the integers from

-7

7

inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?