For The Mathematically Inclined

For The Mathematically Inclined

Anil Saigal
11/24/2008

FOR STUDENTS

PROBLEM # 1
Define x # y = x³ - y. What is h # (h # h)?

FOR GROWNUPS

PROBLEM #2
Compute:
(tan 41^o )(tan 42^o )(tan 43^o )(tan 44^o )(tan 45^o )(tan 46^o )(tan 47^o )(tan 48^o )(tan 49^o ).

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Problem Solutions 11/13/08

PROBLEM # 1
A circle passes through the three vertices of an isoceles triangle that has two sides of length 3 and abase of length 2. Find the area of this circle?

Solution:

7.95 sq. in.

PROBLEM # 2
Note that (2^0.5-1)¹ = 2^0.5 - 1^0.5 and that (2^0.5-1)² = 2 - 2 x 2^0.5 + 1

= 9^0.5 - 8^0.5.

Prove that for all positive integral values of n there exists an integral value of m such that

(2^0.5-1)ⁿ = m^0.5 - (m-1)^0.5.

Solution:

Congratulations to Sitaram Chandawarkar and Shailesh Goregaokar, who were winners of the last set of puzzles.

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