Solution: Let x be the larger number and y the smaller number.

Transcription

1 Problem The sum of two numbers is 00 The lrger number minus the smller number is Find the numbers [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: Vin Lee] Solution: Let be the lrger number nd y the smller number y = 00 y = = = 6 y = 00 6 = 6

2 Problem The rtio of the legs of right tringle is to 4 If the hypotenuse of the right tringle is 0 cm, wht re the mesures of the legs? [Problem submitted by Kee Lm, LACC Professor of Mthemtics Source: Kee Lm] Solution: Let the mesures of the legs of the right tringle re nd 4, then by Pythgoren Theorem, 4 = 0 ( ) ( ) ( ) = 900 = 900 = 6 = 6 Hence, the mesures of the legs of the right tringle re ( 6) = 8cm nd 4 ( 6) = 4cm

3 Problem Find the vlue of if = [Problem submitted by Kee Lm, LACC Professor of Mthemtics Source: Kee Lm] Solution: Cube both sides of the eqution nd then simplify: / / / / = ( ) ( ) ( ) ( ) ( ) ( ) / ( ) / ( ) / ( ) / ( ) ( ) / ( ) / ( ) / ( ) / [ ] ( ) / = = = = 8 = = ± 9

4 Problem 4 The solution of the eqution = 8 cn be epressed in the form = log b Find b [Problem submitted by H Nguyen, LACC Adjunct Instructor of Mthemtics Source: Mthemticl Assocition of Americ s Americn Mthemtics Competition 00 Problem ] Solution: = 8 = 8 8 = = log 8 8 So, b =

5 Problem 5 If = 0, Find [Problem submitted by Munir Smplewl, LACC Professor of Computer Science nd Informtion Technology Source: Munir Smplewl] Solution: = 0 = 000 = 000 = ( ) = 000 = 90

6 Problem 6 A sequence { n } stisfies = n n n for ll n 4 If = nd = 6 0, find 8 [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: AMATYC Student Mthemtics Legue, October/November 04] Solution: Let = nd = y Then 4 = y 5 = y = y 6 = y y = y y y = ( y ) = (0) 8 = 6

7 Problem For the function f (), f ( ) = 4 Also, f ( ) f ( y) = f ( y) f ( y) for ll rel numbers nd y Find f (5) [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: AMATYC Fculty Mthemtics Legue, Test I, November 4, 04] Solution: f ( ) f (0) = f ( 0) f ( 0) 4 f (0) = 4 4 f ( 0) = f ( ) f () = f ( ) f ( ) 4 4 = f () f ( ) = 4 f ( ) f () = f ( ) f ( ) 4 4 = f () 4 f ( ) = 5 f ( ) f () = f ( ) f ( ) 5 4 = f (5) 4 f ( 5) = 4

8 Problem 8 A cubic eqution hs three roots which re perfect squres such tht b = c, where, b, nd c re the three roots If the eqution is p q r = 0, find the reltion tht holds mong p, q, nd r [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: Sint Mry s College Mthemtics Contest Problems by Brother Alfred Brousseu, Cretive Publictions, 9] Solution: Since, b, nd c re the three roots, ( )( b )( c ) = 0 Multiply, collect like terms, nd equte to the given cubic eqution to get ( b c ) ( b c b c ) b c = p q r Equte the coefficients of like terms to get b c = p c = p b c b c = q b c ( b ) = q b c 4 = q b p = q 4 b c = r b p = r b = r p Therefore, r p 4 p = q So, 8r p = 4 pq

9 Problem 9 Find ll prime numbers of the form 00 00, where the number of zeros between the first nd lst digits is even [Problem submitted by Iris Mgee, LACC Professor of Mthemtics Source: Wisconsin Mthemtics Science & Engineering Tlent Serch] Solution: Note tht if m is odd nd positive, then m b m = m m m m 4 m ( b)( b b b b ) Let k be the number of zeros in Then 0000 = 0 k Since k is even, k is odd So, is fctorble s shown bove with one of the fctors being (0) Therefore, is the only prime number of the form in which the number of zeros between the first nd lst digits is even k

10 Problem 0 For the sequence, 8, 09, 4,, -,, n, find the polynomil of lest degree, f(n), such tht n = f (n) for n =,,, [Problem submitted by Vin Lee, LACC Professor of Mthemtics Source: Vin Lee] Solution: For ny sequence,,, 4, define the first difference sequence to be,, 4, 5 4, Define the second difference sequence to be the first difference sequence of the first difference sequence nd so on Consider the sequence defined by n = n b : b, b, b, Its first difference sequence is,,, Net consider the sequence defined by n = n bn c : b c, 4 b c,9 b c, Its first difference sequence is b,5 b c, b, Its second difference sequence is,,, Now consider the sequence defined by n = n bn cn d : b c d, 8 4b c d, 9b c d, Its third difference sequence is 6,6,6, Therefore, if k =,, or for sequence whose terms re defined by kth degree polynomil, the kth difference sequence is constnt sequence whose terms re k! where is the leding coefficient of the polynomil Applying this observtion to the given sequence in this problem,, 8, 09, 4,, -,, the first, second nd third difference sequences re: first: 5, -9, -5, -, -, second: -4, -6, -8, -50, third: -, -, -, This implies the sequence is defined by third degree polynomil: f ( n) = n bn cn d with! = So, = - nd d cn bn = f ( n) n Substitute n =, n =, nd n = into this eqution to get three equtions with three unknowns: d c b = 5 d c 4b = 4 d c 9b = 6 Solve this system to get d = 06, c = 4, nd b = 5 Therefore, n = f ( n) = n 5n 4n 06