1.


2. f(x) = x18-12x15+36x12-2560x9-13104x6-17472x3-64 is one such function.
.......Update: Acknowledgements to correspondent Jeremy Weissmann, who found the following polynomial of degree 6:
.......f(x) = x6-6x4-4x3+12x2-24x-4 .


3. s = 35 ; n = 4 ; k = 9


4. , or phi, also known as the golden ratio or golden section. It is equal to .


5. (sinh 1)/2


6. (sinh 1 - sin 1)/8


7.


8. 11

9. 182


10. 8,018,844,304 (eight billion, eighteen million, eight hundred forty-four thousand, three hundred four), which is the square of 89,548.

11. 2,000,000,000,000,000,000,000,000,002,202,211,698,297,079,027,232,663,249,060,167,009 (two vigintillion, two undecillion, two hundred two decillion, two hundred eleven nonillion, six hundred ninety-eight octillion, two hundred ninety-seven septillion, seventy-nine sextillion, twenty-seven quintillion, two hundred thirty-two quadrillion, six hundred sixty-three trillion, two hundred forty-nine billion, sixty million, one hundred sixty-seven thousand, nine), which is the square of 44,721,359,549,995,793,928,183,473,399,247.


12. "first";"second";"third"


13. e + (sinh 1)/4


14. I found a formula on an internet site,
http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html#twotriangles ,
which aided me in determining the correct solution:
(, ).
Before finding the above formula I had gotten the solution simplified only to the point shown below after many attempts over many years:
Incenter is at (x0, y0), where

x0 =
3 + ,

and y0 =
.

15. a) 2

b) (see #4 above)

c)


d)



16.



17. a) I found expressions for 0, 1, 2, 4, 5, 6, 8, 10, 12, 16, 20, 22, 24, 26, 28, 44, 48, 60, 64, 96, 120, 256, 576, 720, 4096, 13824, 40320, 331776, 3628800, 16777216, 191102976, 479001600, 20922789888000, 281474976710656, 36520347436056576, 2432902008176640000, 1124000727777607680000, 25852016738884976640000, 155112100433309859840000, 248179360693295775744000, 310224200866619719680000, 620448401733239439359976, 620448401733239439359996, 620448401733239439359998, 620448401733239439360000, 620448401733239439360002, 620448401733239439360004, 620448401733239439360024, 1240896803466478878720000, 1551121004333098598400000, 2481793606932957757440000, 14890761641597746544640000, 403291461126605635584000000, 304888344611713860501504000000, 1333735776850284124449081472843776, 384956219213331276939737002152967117209600000000, 2658271574788448768043625811014615890319638528000000000, 12413915592536072670862289047373375038521486354677760000000000, 238845470948181951014676282087587661242764623832882511609856000000000000, 8320987112741390144276341183223364380754172606361245952449277696409600000000000000, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000, and 148191290711022364575836328127140926575194970820976779184863158185815504490332160000000000000000.
Click here to view.

b) I found expressions for 1/64, 1/60, 16/625, 1/24, 1/16, 8/125, 1/12, 1/10, 1/8, 4/25, 1/6, 1/5, 1/4, 2/5, 11/25, 1/2, 4/5, 8/5, 12/5, 5/2, 18/5, 22/5, 25/4, 48/5, 125/8, 118/5, 122/5, and 625/16.
Click here to view.

c)
would work.


d)



18. If the radius of the circle is less than 1, there is no solution. Otherwise,
BP = .
[Update: It turns out there is a standard theorem that addresses this scenario, so evidently the problem is not so difficult as I had earlier believed.]

19. The mean is 333833.5 ; the median is 250500.5 .

20. a) 1/120       b) 1/120       c) 3/40       d) 1/120       e) 1/120       f) 1/120       g) 7/120       h) 1/40       i) 1/120       j) 1/120       k) 7/120       l) 1/40       m) 1/120       n) 1/120       o) 1/120       p) 1/120       q) 1/120       r) 1/120       s) 61/120       t) 17/120       u) infinitesimal

21. 41,038

22. a) 1, 1, 1
xxxb) 1, 4, 4
xxxc) 16, 25, 36
xxxd) no solution (this is easily proven using the Pythagorean Theorem and Fermat's Last Theorem!)
xxxe) 1, 1, 1
xxxf) 9, 9, 16
xxxg) 16, 25, 36
xxxh) 64, 81, 100

23. Since the Hubble length is not defined with great precision, an exact answer is not possible. However, the answer would be roughly on the order of 10385.

24. I counted 11,148.

25.


26. You could say either "Bonjour!" (in French, the official language) or "Bara ala!" (in Sango, the national language) from your position at about 7.2ºN, 21.5ºE in Central African Republic. Your friend would likely reply "Asalaamu Aleikum" (in Arabic), or perhaps "Nabad" (in Somali), from his or her position at approximately 3.6ºN, 43.0ºE in Somalia.

27. 52

28. One such function would be:
,
which includes the points (1, ), (2, ), (3, ), and (4, ); the tangents at these points enclose the square with vertices (, ), (, ), (, ), and (, ), which has area .

29. (see #4 above)

30. a) 62
......b) Perimeter = 961; Angle measures (in radians): 1, 1, and ; Sides: two each measuring 961/(2 + 2 cos 1), and one measuring (961 cos 1)/(1 + cos 1); Area: (923521 sin 2)/(12 + 16 cos 1 + 4 cos 2)

31.  


32.  


33.  


34.  


35.  


36.   35


37. a)   45,360
   b)   50,400 (which has 108 factors)


38.     (2345 out of 9000)


39.  a)  eight billion, eighteen million, eighteenth
   b)  "Eighth" is the 1,001,001,001,001,001,001,001,002,001,002,001,001,001,001,001,001,001,002,001,001,002nd entry on the list.


40.  516


41.  2304


42. a)   276
   b)   Each of two distinct triangles has the maximum area: one with perimeter and one with perimeter   .


43.  688


44.  9n-3(4n2-3n-1)


45.  1 — 0.9 ln 9! + 8.1 ln 10


46.  819199/819200 (which equals exactly 0.999998779296875)


47.  (sin 2°)/2


48. a)  2700
      b)  7


49.  



50.  –3


51.   ,   which can also be expressed as ,   ,   or


52.  3, 5, 9, 15, 16. A very liberal interpretation of the conditions might also allow polygons of 48, 80, 144, and 240 sides. I choose not to consider these, since each would require a 180° angle to be regarded as an interior angle of a polygon. . . . One might also argue that one 180° angle could be counted, but since the "sides" meeting at its vertex would be collinear, they could be combined to make one side, resulting in polygons of 47, 79, 143, and 239 sides possibly satisfying the conditions. (I would still be inclined to reject such an argument.)


53. a) 999,968
      b) 29
      c) 3


54. a) –2
      b) 37,748,736


55. a) 0 < x < 10/3; 0 < y < 5; 10/3 < z < 10
      b) 0 < x < 3; 1/2 < y < 5; 7/2 < z < 9
      c) x = 1, 2, or 3; y = 1, 2, 3, or 4; z = 4, 5, 6, 7, or 8
      d) x = 1, 2, or 3; y = 1, 2, 3, or 4; z = 4, 5, 6, 7, or 8


56.   9/65536


57. Either:
j = k = 1, 2, 3, or 4;
or:
j = 0, 1, 2, or 3; and k is any integer that exceeds j by at least 2.


58.   1061/1250, or .8488


59.  


60. a) 8,222,838,654,177,922,817,725,562,880,000,000 (eight decillion, two hundred twenty-two nonillion, eight hundred thirty-eight octillion, six hundred fifty-four septillion, one hundred seventy-seven sextillion, nine hundred twenty-two quintillion, eight hundred seventeen quadrillion, seven hundred twenty-five trillion, five hundred sixty-two billion, eight hundred eighty million), which is 31!.
      b) 2,658,271,574,788,448,768,043,625,811,014,615,890,319,638,528,000,000,000 (two septendecillion, six hundred fifty-eight sexdecillion, two hundred seventy-one quindecillion, five hundred seventy-four quattuordecillion, seven hundred eighty-eight tredecillion, four hundred forty-eight duodecillion, seven hundred sixty-eight undecillion, forty-three decillion, six hundred twenty-five nonillion, eight hundred eleven octillion, fourteen septillion, six hundred fifteen sextillion, eight hundred ninety quintillion, three hundred nineteen quadrillion, six hundred thirty-eight trillion, five hundred twenty-eight billion), which is 44!.


61. a) 300 (three hundred, which spelled backwards is derdnuh eerht).
      b) 660,000,000,000,000,000,000,000,000,000,000,000,000,060,060,000,000,000,000,000,000,060 (six hundred sixty vigintillion, sixty septillion, sixty sextillion, sixty, which spelled backwards is ytxis noillitxes ytxis noillitpes ytxis noillitnigiv ytxis derdnuh xis).


62.    166


63.    242


64.    (, )   (i.e., there are two such points)


65. a) 111,111,111,113
      b) 617,286
      c) 54,869,686
      d) 44,544,551
      e) 11,111,111,111,111,111,111,111,111,122,222,222,222,222,222,222,222,233,333,333,348
      f) 64
      g) n = 5k2+2
      h) 542,526


66.    8,281


67. a) 100
      b) 0


68. a) 4,342,858,240
      b) 6


69.    1107/2000, or .5535


70.    (1, 25, 269, 2789, 27999, 289999, 2999999, 37999999, 379999999, 3899999999, 39999999999, 489999999999, 4899999999999, 49999999999999, 589999999999999, 5999999999999999, 69999999999999999, 699999999999999999, 7999999999999999999, 89999999999999999999, 999999999999999999999)


71. a) 18,888,888,888,888,888,888,888,888
      b) 13 different values of n, the smallest of which is 37


72. a) 12 hr, 2 2/3 min (i.e., exactly 12:02:40) . . . 12 hr, 53 7/13 min (approximately 12:53:32) . . . 1 hr, 4 6/13 min (approximately 1:04:28)
      b) 1 hr, 4 8/13 min (approximately 1:04:37) . . . 11 hr, 5 5/11 min (approximately 11:05:27) (Interestingly, for the latter question, the angle between the hands would be exactly 60 degrees.)


73.   9. They are:
   {36}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 18}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 12}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 9}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 9}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 4}
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3}


74. a) 1/5       b) 1/20       c) 1/20       d) 1/10       e) 1/10       f) 1/5       g) 1/20       h) 1/20       i) 1/10       j) 1/10      


75. I counted 1,376.


76. a) the 134-digit number represented as a 2, followed by 100 5s, followed by 33 8s
      b) the 105-digit number represented as a 6 followed by 104 9s


77. a) 51,515,158,585
      b) 1,515,151,515,151,212,121,212,121


78.     698,989,898,989


79.     109


80. a) 20,200
      b) 1,202,100


81. a) 25,569
      b) 5899


82.     15,710,112




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