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Maths Questions
1. Let t>1 a) Show that t^2 /t-1 ≧ 4 x,y,z ≧ 1 b) x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c) x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48 2. Prove that 5^n - 25 is divisible by 100 for n ≧ 2. How can I get the answer? Thank a lot!
最佳解答:
Q1: (a) t^2/(t-1)=(t+1) + 1/(t-1)= 2+ (t-1)+1/(t-1) >= 2 + 2=4 Without loss of generality, we can suppose the x>=y>=z>=1 for (b)and(c) (b) x^2/(y-1) + y^2/(z-1) + z^2/(x-1) - [y^2/(y-1) + z^2/(z-1) + x^2/(x-1) ] =(x^2-y^2)/(y-1)+ (y^2-z^2)/(z-1) - [(x^2-y^2)+(y^2-z^2)]/(x-1) =(x^2-y^2)[1/(y-1) - 1/(x-1) ] + (y^2-z^2)[ 1/(z-1) - 1/(x-1)] >= 0 (Since x^2>=y^2>=z^2 and 1/(z-1)>=1/(y-1)>=1/(x-1) ) thus, x^2/(y-1)+..+z^2/(x-1) >= y^2/(y-1)+z^2/(z-1)+x^2/(x-1) (by (a) ) >= 4+4+4=12 (c) x^4/(y-1)^2+y^4/(z-1)^2+ z^4/(x-1)^2 -[y^4/(y-1)^2+z^4/(z-1)^2+x^4/(x-1)^2] =(x^4-y^4)/(y-1)^2+ (y^4-z^4)/(z-1)^2 -[(x^4-y^4)+(y^4-z^4)]/(x-1)^2 =(x^4-y^4)[1/(y-1)^2 - 1/(x-1)^2 ] + (y^4-z^4)[ 1/(z-1)^2 - 1/(x-1)^2] >= 0 (Since x^4>=y^4>=z^4 and 1/(z-1)^2>=1/(y-1)^2>=1/(x-1)^2 ) thus, x^4/(y-1)^2+..+z^4/(x-1)^2 >= y^4/(y-1)^2+z^4/(z-1)^2+x^4/(x-1)^4 (by (a) ) >= 4^2+4^2+4^2=48 Note: the results can be generalized. Q2: (1) Obviously 25 | (5^n-25) (2) 5^n = 1 (mod 4), so 5^n - 25 = 1- 1 (mod 4) so that 4*25 | (5^n - 25) (Alternative method) Let n=2+k, then 5^n-25=25[5^k -1]=25*(5-1)[5^(k-1)+5^(k-2)+...+5+1] =100*A so, 5^n - 25 is divisible by 100.
其他解答:
1a) t>1 t-2>-1 (t-1)^2≧0 t^2-4t+4≧0 t^2≧4t-4 t^2 /t-1 ≧ 4 (t-1>0) b)By(a)x^2 / y-1,y^2 /z-1,z^2 / x-1≧ 4 so,x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c)(x^2 / y-1)^2,(y^2 /z-1)^2,(z^2 / x-1)^2≧ 16 so,x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48 2)let P(n) be '5^n - 25 is divisible by 100 for n ≧ 2' for n=2 5^2-25=0 which is divisible by 100 assume P(k) is true,5^k-25=100M for n=k+1 5^(k+1)-25=5x5^k-25 =5(100M+25)-25=500M+100 which is divisible by 100 P(k+1) is true 2010-02-21 09:33:25 補充： 1a) t>1 t-2>-1 (t-1)^2≧0 t^2-4t+4≧0 t^2≧4t-4 t^2 /t-1 ≧ 4 (t-1>0) b)By(a)x^2 / y-1,y^2 /z-1,z^2 / x-1≧ 4 so,x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c)(x^2 / y-1)^2,(y^2 /z-1)^2,(z^2 / x-1)^2≧ 16 so,x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48
Maths Questions
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發問:1. Let t>1 a) Show that t^2 /t-1 ≧ 4 x,y,z ≧ 1 b) x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c) x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48 2. Prove that 5^n - 25 is divisible by 100 for n ≧ 2. How can I get the answer? Thank a lot!
最佳解答:
Q1: (a) t^2/(t-1)=(t+1) + 1/(t-1)= 2+ (t-1)+1/(t-1) >= 2 + 2=4 Without loss of generality, we can suppose the x>=y>=z>=1 for (b)and(c) (b) x^2/(y-1) + y^2/(z-1) + z^2/(x-1) - [y^2/(y-1) + z^2/(z-1) + x^2/(x-1) ] =(x^2-y^2)/(y-1)+ (y^2-z^2)/(z-1) - [(x^2-y^2)+(y^2-z^2)]/(x-1) =(x^2-y^2)[1/(y-1) - 1/(x-1) ] + (y^2-z^2)[ 1/(z-1) - 1/(x-1)] >= 0 (Since x^2>=y^2>=z^2 and 1/(z-1)>=1/(y-1)>=1/(x-1) ) thus, x^2/(y-1)+..+z^2/(x-1) >= y^2/(y-1)+z^2/(z-1)+x^2/(x-1) (by (a) ) >= 4+4+4=12 (c) x^4/(y-1)^2+y^4/(z-1)^2+ z^4/(x-1)^2 -[y^4/(y-1)^2+z^4/(z-1)^2+x^4/(x-1)^2] =(x^4-y^4)/(y-1)^2+ (y^4-z^4)/(z-1)^2 -[(x^4-y^4)+(y^4-z^4)]/(x-1)^2 =(x^4-y^4)[1/(y-1)^2 - 1/(x-1)^2 ] + (y^4-z^4)[ 1/(z-1)^2 - 1/(x-1)^2] >= 0 (Since x^4>=y^4>=z^4 and 1/(z-1)^2>=1/(y-1)^2>=1/(x-1)^2 ) thus, x^4/(y-1)^2+..+z^4/(x-1)^2 >= y^4/(y-1)^2+z^4/(z-1)^2+x^4/(x-1)^4 (by (a) ) >= 4^2+4^2+4^2=48 Note: the results can be generalized. Q2: (1) Obviously 25 | (5^n-25) (2) 5^n = 1 (mod 4), so 5^n - 25 = 1- 1 (mod 4) so that 4*25 | (5^n - 25) (Alternative method) Let n=2+k, then 5^n-25=25[5^k -1]=25*(5-1)[5^(k-1)+5^(k-2)+...+5+1] =100*A so, 5^n - 25 is divisible by 100.
其他解答:
1a) t>1 t-2>-1 (t-1)^2≧0 t^2-4t+4≧0 t^2≧4t-4 t^2 /t-1 ≧ 4 (t-1>0) b)By(a)x^2 / y-1,y^2 /z-1,z^2 / x-1≧ 4 so,x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c)(x^2 / y-1)^2,(y^2 /z-1)^2,(z^2 / x-1)^2≧ 16 so,x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48 2)let P(n) be '5^n - 25 is divisible by 100 for n ≧ 2' for n=2 5^2-25=0 which is divisible by 100 assume P(k) is true,5^k-25=100M for n=k+1 5^(k+1)-25=5x5^k-25 =5(100M+25)-25=500M+100 which is divisible by 100 P(k+1) is true 2010-02-21 09:33:25 補充： 1a) t>1 t-2>-1 (t-1)^2≧0 t^2-4t+4≧0 t^2≧4t-4 t^2 /t-1 ≧ 4 (t-1>0) b)By(a)x^2 / y-1,y^2 /z-1,z^2 / x-1≧ 4 so,x^2 / y-1 + y^2 /z-1 + 2^2 / x-1 ≧ 12 c)(x^2 / y-1)^2,(y^2 /z-1)^2,(z^2 / x-1)^2≧ 16 so,x^4 / (y-1)^2 + y^4 / (z-1)^2 + z^2 / (x-1)^2 ≧ 48
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