Statistics

Problem Statement for "QuadraticEquations"

Problem Statement

Young Andrew has just learned about quadratic equations. He was quite amazed by the fact that their solutions could look like (5+sqrt(3))/4, so he wants to dig into this issue. More specifically, given five numbers x, y, d, z and k, he wants to find the number of equations a*t2+b*t+c=0 such that (x+y*sqrt(d))/z is a solution of the equation (i.e., when substituting it for t the equation holds) and a, b and c are integers, -k <= a, b, c <= k. Notice that the equations he's looking for are not necessarily quadratic, i.e., a is allowed to be zero, as is b and/or c.

Definition

Class:
QuadraticEquations
Method:
howMany
Parameters:
int, int, int, int, int
Returns:
long
Method signature:
long howMany(int x, int y, int d, int z, int k)
(be sure your method is public)

Constraints

  • x, y and z will be between -1000 and 1000, inclusive.
  • z will not be equal to 0.
  • d will be between 1 and 1000, inclusive.
  • k will be between 0 and 1000000 (106), inclusive.

Examples

  1. 5

    1

    3

    4

    30

    Returns: 3

    The three possible equations are: 0*x2+0*x+0=0, 8*x2-20*x+11=0, -8*x2+20*x-11=0

  2. 5

    1

    3

    4

    10

    Returns: 1

    The equation 0*x2+0*x+0=0 always holds.

  3. 2

    0

    1

    1

    2

    Returns: 7

    The value described is simply 2. The equations are 0*x2+0*x+0=0, 0*x2+1*x-2=0, 0*x2-1*x+2=0, 1*x2-1*x-2=0, -1*x2+1*x+2=0, 1*x2-2*x+0=0, -1*x2+2*x+0=0.

  4. 0

    -1

    4

    -1

    2

    Returns: 7

    It is still 2.

  5. -1

    3

    3

    2

    1000000

    Returns: 153847

  6. 2

    0

    1

    3

    5

    Returns: 11

  7. 0

    1

    2

    1000

    574891

    Returns: 3

  8. 1

    1

    2

    1000

    454739

    Returns: 1

  9. 1

    1

    2

    1000

    0

    Returns: 1

  10. 1

    -1

    2

    1000

    480019

    Returns: 1

  11. 0

    1

    2

    100

    574891

    Returns: 229

  12. 1

    1

    2

    100

    454739

    Returns: 91

  13. 1

    1

    2

    100

    0

    Returns: 1

  14. 1

    -1

    2

    100

    480019

    Returns: 97

  15. 0

    0

    1

    57

    1000000

    Returns: 4000004000001

  16. 0

    0

    1

    1

    0

    Returns: 1

  17. 5

    -1

    25

    999

    968900

    Returns: 3755072715601

  18. -1000

    -1000

    1000

    1

    459595

    Returns: 1

  19. 1

    0

    1

    1000

    224116

    Returns: 201385

  20. 0

    -5

    4

    3

    43665

    Returns: 76266163

  21. 12

    -5

    49

    9

    75185

    Returns: 42738053

  22. 64

    -437

    208

    850

    1000000

    Returns: 1

  23. 108

    -467

    96

    569

    1000000

    Returns: 1

  24. -966

    -286

    686

    661

    1000000

    Returns: 1

  25. 0

    -5

    9

    2

    1

    Returns: 1

  26. -35

    46

    16

    96

    149

    Returns: 7

  27. -49

    -48

    8

    -88

    16031

    Returns: 3

  28. -733

    672

    169

    790

    27

    Returns: 1

  29. 4

    -39

    76

    -10

    741758

    Returns: 257

  30. -996

    -746

    211

    775

    6238

    Returns: 1

  31. -6

    0

    1

    -8

    670279

    Returns: 105818531049

  32. -603

    734

    841

    -341

    267855

    Returns: 643

  33. -10

    0

    9

    7

    411005

    Returns: 6579236979

  34. 53

    19

    99

    -44

    197580

    Returns: 25

  35. 9

    0

    6

    10

    252

    Returns: 2145

  36. 98

    -8

    17

    -14

    826052

    Returns: 777

  37. 531

    -114

    784

    969

    694057

    Returns: 2449157

  38. -842

    -906

    409

    533

    1000000

    Returns: 1

  39. 6

    -1

    7

    -9

    6804

    Returns: 127

  40. -2

    2

    5

    -4

    247791

    Returns: 495583

  41. 9

    2

    6

    8

    961

    Returns: 13

  42. 731

    -767

    256

    327

    788390

    Returns: 167641

  43. -469

    497

    961

    855

    14937

    Returns: 1

  44. 530

    195

    484

    245

    355484

    Returns: 543555

  45. 1

    -5

    3

    1

    73

    Returns: 1

  46. 95

    -16

    53

    -34

    19379

    Returns: 5

  47. -12

    37

    12

    -51

    569940

    Returns: 211

  48. 26

    35

    100

    -53

    928232

    Returns: 24375979

  49. -8

    -4

    9

    5

    1787

    Returns: 798119

  50. -6

    0

    9

    -3

    1

    Returns: 1

  51. 499

    -841

    875

    -170

    1000000

    Returns: 1

  52. 97

    35

    6

    71

    440186

    Returns: 4539

  53. 47

    -42

    64

    87

    203064

    Returns: 1974433

  54. 769

    -962

    361

    -218

    536239

    Returns: 3723

  55. 9

    2

    1

    4

    799483

    Returns: 21129767463

  56. 84

    37

    81

    -64

    416

    Returns: 1

  57. 19

    -15

    76

    -45

    870427

    Returns: 103

  58. -898

    229

    841

    -754

    580495

    Returns: 41039

  59. -4

    4

    7

    -9

    3

    Returns: 1

  60. -444

    232

    256

    -356

    670631

    Returns: 2694017

  61. 276

    946

    255

    -883

    94

    Returns: 1

  62. 60

    -5

    267

    -928

    861184

    Returns: 3

  63. -449

    756

    576

    250

    7027

    Returns: 9

  64. 76

    17

    31

    100

    3215

    Returns: 1

  65. -9

    0

    3

    -9

    546383

    Returns: 895604787217

  66. -148

    -531

    163

    -685

    1000000

    Returns: 1

  67. -3

    -2

    3

    2

    767084

    Returns: 127847

  68. -7

    -3

    4

    6

    386059

    Returns: 3527574135

  69. -343

    -96

    625

    -780

    103762

    Returns: 966809

  70. 0

    1

    2

    1

    575757

    Returns: 575757

  71. 0

    1

    3

    -1

    1000000

    Returns: 666667

  72. 0

    0

    2

    1

    1000000

    Returns: 4000004000001

  73. 1

    4

    9

    49

    100000

    Returns: 16657225

  74. 0

    0

    2

    1

    1000

    Returns: 4004001

  75. 4

    -4

    1

    10

    100

    Returns: 40401

  76. 1

    5

    25

    39

    1000000

    Returns: 439815296297

  77. -1

    3

    3

    2

    1000000

    Returns: 153847

  78. 1

    1

    1

    3

    1000000

    Returns: 439815296297

  79. 0

    0

    1

    1

    1000000

    Returns: 4000004000001

  80. -100

    -233

    121

    -99

    1000000

    Returns: 564025

  81. 1

    -1

    1

    1

    1000000

    Returns: 4000004000001

  82. 5

    0

    3

    2

    77

    Returns: 961

  83. 17

    3

    19

    14

    89528

    Returns: 753

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