11 if a 1 a 2 a n are all positive and s n a 1 a 2 a

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an inequality usually known as Schwarz’s (though due originally to Cauchy). 11. If a 1 , a 2 , . . . , a n are all positive, and s n = a 1 + a 2 + · · · + a n , then (1 + a 1 )(1 + a 2 ) . . . (1 + a n ) 5 1 + s n + s 2 n 2! + · · · + s n n n ! . ( Math. Trip. 1909.) 12. If a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n are two sets of positive numbers, arranged in descending order of magnitude, then ( a 1 + a 2 + · · · + a n )( b 1 + b 2 + · · · + b n ) 5 n ( a 1 b 1 + a 2 b 2 + · · · + a n b n ) . 13. If a , b , c , . . . k and A , B , C , . . . K are two sets of numbers, and all of the first set are positive, then aA + bB + · · · + kK a + b + · · · + k lies between the algebraically least and greatest of A , B , . . . , K . 14. If p , q are dissimilar surds, and a + b p + c q + d pq = 0, where a , b , c , d are rational, then a = 0, b = 0, c = 0, d = 0. [Express p in the form M + N q , where M and N are rational, and apply the theorem of § 14 .] 15. Show that if a 2 + b 3 + c 5 = 0, where a , b , c are rational numbers, then a = 0, b = 0, c = 0. 16. Any polynomial in p and q , with rational coefficients ( i.e. any sum of a finite number of terms of the form A ( p ) m ( q ) n , where m and n are integers, and A rational), can be expressed in the form a + b p + c q + d pq,
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[I : 19] REAL VARIABLES 38 where a , b , c , d are rational. 17. Express a + b p + c q d + e p + f q , where a , b , etc. are rational, in the form A + B p + C q + D pq, where A , B , C , D are rational. [Evidently a + b p + c q d + e p + f q = ( a + b p + c q )( d + e p - f q ) ( d + e p ) 2 - f 2 q = α + β p + γ q + δ pq + ζ p , where α , β , etc. are rational numbers which can easily be found. The required reduction may now be easily completed by multiplication of numerator and denominator by - ζ p . For example, prove that 1 1 + 2 + 3 = 1 2 + 1 4 2 - 1 4 6 . ] 18. If a , b , x , y are rational numbers such that ( ay - bx ) 2 + 4( a - x )( b - y ) = 0 , then either (i) x = a , y = b or (ii) 1 - ab and 1 - xy are squares of rational numbers. ( Math. Trip. 1903.) 19. If all the values of x and y given by ax 2 + 2 hxy + by 2 = 1 , a 0 x 2 + 2 h 0 xy + b 0 y 2 = 1 (where a , h , b , a 0 , h 0 , b 0 are rational) are rational, then ( h - h 0 ) 2 - ( a - a 0 )( b - b 0 ) , ( ab 0 - a 0 b ) 2 + 4( ah 0 - a 0 h )( bh 0 - b 0 h ) are both squares of rational numbers. ( Math. Trip. 1899.) 20. Show that 2 and 3 are cubic functions of 2 + 3, with rational coefficients, and that 2 - 6+3 is the ratio of two linear functions of 2+ 3. ( Math. Trip. 1905.)
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[I : 19] REAL VARIABLES 39 21. The expression q a + 2 m p a - m 2 + q a - 2 m p a - m 2 is equal to 2 m if 2 m 2 > a > m 2 , and to 2 a - m 2 if a > 2 m 2 . 22. Show that any polynomial in 3 2, with rational coefficients, can be ex- pressed in the form a + b 3 2 + c 3 4 , where a , b , c are rational. More generally, if p is any rational number, any polynomial in m p with rational coefficients can be expressed in the form a 0 + a 1 α + a 2 α 2 + · · · + a m - 1 α m - 1 , where a 0 , a 1 , . . . are rational and α = m p . For any such polynomial is of the form b 0 + b 1 α + b 2 α 2 + · · · + b k α k , where the b ’s are rational. If k 5 m - 1, this is already of the form required. If k > m - 1, let α r be any power of α higher than the ( m - 1)th. Then r = λm + s , where λ is an integer and 0 5 s 5 m - 1; and α r = α λm + s = p λ α s . Hence we can get rid of all powers of α higher than the ( m - 1)th.
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