mathematics from hungary as they prepare for olympiads

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Mathematics from Hungary as they prepare for Olympiads

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  • Contributors

    Chan Kin Hang, 1998, 1999, 2000, 2001 Hong Kong team member

    Chan Ming Chiu, 1997 Hong Kong team reserve member

    Chao Khek Lun, 2001 Hong Kong team member

    Cheng Kei Tsi, 2001 Hong Kong team member

    Cheung Pok Man, 1997, 1998 Hong Kong team member

    Fan Wai Tong, 2000 Hong Kong team member

    Fung Ho Yin, 1997 Hong Kong team reserve member

    Ho Wing Yip, 1994, 1995, 1996 Hong Kong team member

    Kee Wing Tao, 1997 Hong Kong team reserve member

    Lam Po Leung, 1999 Hong Kong team reserve member

    Lam Pei Fung, 1992 Hong Kong team member

    Lau Lap Ming, 1997, 1998 Hong Kong team member

    Law Ka Ho, 1998, 1999, 2000 Hong Kong team member

    Law Siu Lung, 1996 Hong Kong team member

    Lee Tak Wing, 1993 Hong Kong team reserve member

    Leung Wai Ying, 2001 Hong Kong team member

    Leung Wing Chung, 1997, 1998 Hong Kong team member

    Mok Tze Tao, 1995, 1996, 1997 Hong Kong team member

    Ng Ka Man, 1997 Hong Kong team reserve member

    Ng Ka Wing, 1999, 2000 Hong Kong team member

    Poon Wai Hoi, 1994, 1995, 1996 Hong Kong team member

    Poon Wing Chi, 1997 Hong Kong team reserve member

    Tam Siu Lung, 1999 Hong Kong team reserve member

    To Kar Keung, 1991, 1992 Hong Kong team member

    Wong Chun Wai, 1999, 2000 Hong Kong team member

    Wong Him Ting, 1994, 1995 Hong Kong team member

    Yu Ka Chun, 1997 Hong Kong team member

    Yung Fai, 1993 Hong Kong team member

    ix

  • Problems

  • Algebra Problems

    Polynomials

    1. (Crux Mathematicorum, Problem 7) Find (without calculus) a fth

    degree polynomial p(x) such that p(x) + 1 is divisible by (x 1)

    3

    and

    p(x) 1 is divisible by (x+ 1)

    3

    :

    2. A polynomial P (x) of the n-th degree satises P (k) = 2

    k

    for k =

    0; 1; 2; : : : ; n: Find the value of P (n + 1):

    3. (1999 Putnam Exam) Let P (x) be a polynomial with real coecients

    such that P (x) 0 for every real x: Prove that

    P (x) = f

    1

    (x)

    2

    + f

    2

    (x)

    2

    + + f

    n

    (x)

    2

    for some polynomials f

    1

    (x); f

    2

    (x); : : : ; f

    n

    (x) with real coecients.

    4. (1995 Russian Math Olympiad) Is it possible to nd three quadratic

    polynomials f(x); g(x); h(x) such that the equation f(g(h(x))) = 0 has

    the eight roots 1; 2; 3; 4; 5; 6; 7; 8?

    5. (1968 Putnam Exam) Determine all polynomials whose coecients are

    all 1 that have only real roots.

    6. (1990 Putnam Exam) Is there an innite sequence a

    0

    ; a

    1

    ; a

    2

    ; : : : of

    nonzero real numbers such that for n = 1; 2; 3; : : : ; the polynomial

    P

    n

    (x) = a

    0

    +a

    1

    x+a

    2

    x

    2

    + +a

    n

    x

    n

    has exactly n distinct real roots?

    7. (1991 Austrian-Polish Math Competition) Let P (x) be a polynomial

    with real coecients such that P (x) 0 for 0 x 1: Show that

    there are polynomials A(x); B(x); C(x) with real coecients such that

    (a) A(x) 0; B(x) 0; C(x) 0 for all real x and

    (b) P (x) = A(x) + xB(x) + (1 x)C(x) for all real x:

    (For example, if P (x) = x(1x); then P (x) = 0+x(1x)

    2

    +(1x)x

    2

    :)

    1

  • 8. (1993 IMO) Let f(x) = x

    n

    + 5x

    n1

    + 3; where n > 1 is an integer.

    Prove that f(x) cannot be expressed as a product of two polynomials,

    each has integer coecients and degree at least 1.

    9. Prove that if the integer a is not divisible by 5, then f(x) = x

    5

    x+ a

    cannot be factored as the product of two nonconstant polynomials with

    integer coecients.

    10. (1991 Soviet Math Olympiad) Given 2n distinct numbers a

    1

    ; a

    2

    ; : : : ; a

    n

    ;

    b

    1

    ; b

    2

    ; : : : ; b

    n

    ; an nn table is lled as follows: into the cell in the i-th

    row and j-th column is written the number a

    i

    + b

    j

    : Prove that if the

    product of each column is the same, then also the product of each row

    is the same.

    11. Let a

    1

    ; a

    2

    ; : : : ; a

    n

    and b

    1

    ; b

    2

    ; : : : ; b

    n

    be two distinct collections of n pos-

    itive integers, where each collection may contain repetitions. If the two

    collections of integers a

    i

    +a

    j

    (1 i < j n) and b

    i

    +b

    j

    (1 i < j n)

    are the same, then show that n is a power of 2.

    Recurrence Relations

    12. The sequence x

    n

    is dened by

    x

    1

    = 2; x

    n+1

    =

    2 + x

    n

    1 2x

    n

    ; n = 1; 2; 3; : : : :

    Prove that x

    n

    6=

    1

    2

    or 0 for all n and the terms of the sequence are all

    distinct.

    13. (1988 Nanchang City Math Competition) Dene a

    1

    = 1; a

    2

    = 7 and

    a

    n+2

    =

    a

    2

    n+1

    1

    a

    n

    for positive integer n: Prove that 9a

    n

    a

    n+1

    + 1 is a

    perfect square for every positive integer n:

    14. (Proposed by Bulgaria for 1988 IMO) Dene a

    0

    = 0; a

    1

    = 1 and a

    n

    =

    2a

    n1

    +a

    n2

    for n > 1: Show that for positive integer k; a

    n

    is divisible

    by 2

    k

    if and only if n is divisible by 2

    k

    :

    2

  • 15. (American Mathematical Monthly, Problem E2998) Let x and y be

    distinct complex numbers such that

    x

    n

    y

    n

    x y

    is an integer for some

    four consecutive positive integers n: Show that

    x

    n

    y

    n

    x y

    is an integer

    for all positive integers n:

    Inequalities

    16. For real numbers a

    1

    ; a

    2

    ; a

    3

    ; : : : ; if a

    n1

    + a

    n+1

    2a

    n

    for n = 2; 3; : : : ;

    then prove that

    A

    n1

    + A

    n+1

    2A

    n

    for n = 2; 3; : : : ;

    where A

    n

    is the average of a

    1

    ; a

    2

    ; : : : ; a

    n

    :

    17. Let a; b; c > 0 and abc 1: Prove that

    a

    c

    +

    b

    a

    +

    c

    b

    a+ b+ c:

    18. (1982 Moscow Math Olympiad) Use the identity 1

    3

    + 2

    3

    + + n

    3

    =

    n

    2

    (n + 1)

    2

    2

    to prove that for distinct positive integers a

    1

    ; a

    2

    ; : : : ; a

    n

    ;

    (a

    7

    1

    + a

    7

    2

    + + a

    7

    n

    ) + (a

    5

    1

    + a

    5

    2

    + + a

    5

    n

    ) 2(a

    3

    1

    + a

    3

    2

    + + a

    3

    n

    )

    2

    :

    Can equality occur?

    19. (1997 IMO shortlisted problem) Let a

    1

    a

    n

    a

    n+1

    = 0 be a

    sequence of real numbers. Prove that

    v

    u

    u

    t

    n

    X

    k=1

    a

    k

    n

    X

    k=1

    p

    k(

    p

    a

    k

    p

    a

    k+1

    ):

    3

  • 20. (1994 Chinese Team Selection Test) For 0 a b c d e and

    a+ b+ c+ d+ e = 1; show that

    ad+ dc+ cb+ be+ ea

    1

    5

    :

    21. (1985 Wuhu City Math Competition) Let x; y; z be real numbers such

    that x + y + z = 0: Show that

    6(x

    3

    + y

    3

    + z

    3

    )

    2

    (x

    2

    + y

    2

    + z

    2

    )

    3

    :

    22. (1999 IMO) Let n be a xed integer, with n 2:

    (a) Determine the least constant C such that the inequality

    X

    1i

  • 26. (Proposed by USA for 1993 IMO) Prove that for positive real numbers

    a; b; c; d;

    a

    b+ 2c+ 3d

    +

    b

    c+ 2d+ 3a

    +

    c

    d+ 2a+ 3b

    +

    d

    a+ 2b+ 3c

    2

    3

    :

    27. Let a

    1

    ; a

    2

    ; : : : ; a

    n

    and b

    1

    ; b

    2

    ; : : : ; b

    n

    be 2n positive real numbers such

    that

    (a) a

    1

    a

    2

    a

    n

    and

    (b) b

    1

    b

    2

    b

    k

    a

    1

    a

    2

    a

    k

    for all k; 1 k n:

    Show that b

    1

    + b

    2

    + + b

    n

    a

    1

    + a

    2

    + + a

    n

    :

    28. (Proposed by Greece for 1987 IMO) Let a; b; c > 0 and m be a positive

    integer, prove that

    a

    m

    b+ c

    +

    b

    m

    c+ a

    +

    c

    m

    a+ b

    3

    2

    a+ b+ c

    3

    m1

    :

    29. Let a

    1

    ; a

    2

    ; : : : ; a

    n

    be distinct positive integers, show that

    a

    1

    2

    +

    a

    2

    8

    + +

    a

    n

    n2

    n

    1

    1

    2

    n

    :

    30. (1982 West German Math Olympiad) If a

    1

    ; a

    2

    ; : : : ; a

    n

    > 0 and a =

    a

    1

    + a

    2

    + + a

    n

    ; then show that

    n

    X

    i=1

    a

    i

    2a a

    i

    n

    2n 1

    :

    31. Prove that if a; b; c > 0; then

    a

    3

    b+ c

    +

    b

    3

    c+ a

    +

    c

    3

    a+ b

    a

    2

    + b

    2

    + c

    2

    2

    :

    32. Let a; b; c; d > 0 and

    1

    1 + a

    4

    +

    1

    1 + b

    4

    +

    1

    1 + c

    4

    +

    1

    1 + d