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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1635.0. "1992 American Invitational Math Exam" by BEING::EDP (Always mount a scratch monkey.) Wed Jul 01 1992 12:30

    Here are the last three.
    
    
    				-- edp
    
    
    13. Triangle ABC has AB=9 and BC:CA=40:41.  What is the largest area
    that this triangle can have?
    
    14. In triangle ABC, A', B', and C' are on sides BC, AC, and AB,
    respectively.  Given that AA', BB', and CC' are concurrent at the point
    O, and that
    
    	AO   BO   CO
    	-- + -- + -- = 92,
    	OA'  OB'  OC'
    
    find the value of
    
    	AO   BO   CO
    	-- * -- * --.
    	OA'  OB'  OC'
    
    15. Define a positive integer n to be a "factorial tail" if there is
    some positive integer m such that the base-ten representation of m!
    ends with exactly n zeroes.  How many positives integers less than 1992
    are not factorial tails?
T.RTitleUserPersonal
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1635.1by expressing area as f of x, and make derivative=0 to find xSTAR::ABBASIi^(-i) = SQRT(exp(PI))Wed Jul 01 1992 14:475
    > 13. Triangle ABC has AB=9 and BC:CA=40:41.  What is the largest area
    >    that this triangle can have?
    
     820 square units ?
    
1635.214DESIR::BUCHANANMon Jul 06 1992 09:1514
14.

    	AO
Let a =	--, and similarly define b & c.
    	OA

    Then by manipulating the obvious vector equations, to eliminate everything
except a,b & c, we can't fail to end up with the pretty:

	abc = a+b+c + 2

whence the answer.

Andrew.