            ⥬ ᫥  䬥᪨ .

                             1.
     ᫮   ᢮  ।⠢ ( ᫠ 
ॢ蠥 10000  ࠧ冷). 室 ।  
 ᫮  15.

                             2.
      ᫮  K-筮 ⥬ ᫥
            a a   ...a   (K<=36).
             n n-1    0
 ⮪     m.
     ᫠ K,  n, m,   ⮪    m, ।⠢
 筮 ⥬ ᫥.

                             3.
      ࠫ쭮  ᫮  N      ⢥   ᯮᮡ
।⠢      楫 ⥫ d[0], ... ,
d[s]  
            N=d[s]*(s+1)!+d[s-1]*s! +...+d[1]*2!+d[0]           (*)
 ᫮,  0<=d[i]<=i+1, i=0,..,s,  d[s]<>0.
      s+1 ࠫ쭮 ᫮ d[0],  ..., d[s],  ࠫ쭮 K,
s<200,d[i]<65535, K<65535.  ⮪   ᫠ N, -
塞  䠪ਠ쭮 ⥬ (*) ᫠ d[0],  ..., d[s], 
᫮ K.

                             4.
     ᫠  U[1],  U[2], ... । 砫묨 -
祭ﬨ U[1]=1, U[2]=2  ᮮ⭮襭
                     U[N+1] = U[N] + U[N-1].
     ᬮਬ ⥬ ᫥   ࠬ 0  1,    -
ன,   ⫨稥    筮  ⥬  ᠬ   ⥯
 1,2,4,8,16,...,   ᫠  1,2,3,5,8,13,....  ⮩
⥬ ᫥    ⥫쭮  楫 ᫮ ⢥
ࠧ ।⠢   ப 㫥  ,  -
  1   ன   ,  冷.
       ப,  ।⠢騥 ᫠ A  B.    ப,
।⠢ ᫮ A+B.
     ਬ. 室 ப '10101'   '100' ।⠢  ᫠
8+3+1=12   3.  ⢥⮬   ப '100010',  ।⠢
ப 13+2=15=12+3.
     ਬ砭. ப    ⮫ ,   ᫠ A  B
ॢ ᨬ쭮 ⨬  襬  楫 ᫮.

                             5.
      ⢮   筮  ᫠ i.

                             6.
     ᫥⥫쭮 011212201220200112... ந ⠪: ᭠-
  0, ⥬  ᫥饥 ⢨: 㦥 ᠭ
 ਯ뢠 ࠢ   0  1, 1  2, 2  0, ..
                0 -> 01 -> 0112 -> 01121220 ->...
     ⠢ ,    N, (0<=N<=3000000000)
।,  ᫮ ⮨  N-   ᫥⥫쭮.

                             7.
      ᨢ  X(100)    Y(100).    ,  騩
᫥⥫쭮 ⠬ 祭 ⮢ X(k)   Y(k)    
⠡,  k=1,2,...,100,  ᯮ ஬ ६.

                             8.
     窨  楫᫥묨 न⠬  1- ࠭ -
 ᫠ 0,1,2,...  ᫥ ࠢ  ᭨  ⠪ ࠧ,
 । 窥 ਯ뢠 쭮  ᫮,饥
  ⨪  ਧ⠫,室饩 १ .  ࢮ -
 窠 (0,0).
      ணࠬ,  
     1.   न⠬ x  y, x>=0, y>=0, x,y- 楫, -
।  窨.
     2.   न x  ⪥ 窨 y, x>=0, y>=0, x,
y - 楫, ।  न 窨.

                             9.
        ਮ  ᠬ ਮ ᪮筮 ⥯ ஡
 ᭮ ,  ।⠢饩 樮쭮 ᫮ N/M ( -
 ஡  ,      ਮ ࠢ 1).  M,N,P - 楫
 ᫠, 0<N<M, 1<P.

                            9.1.
        ⢨⥫쭮 ᫠ r>0  ࠫ쭮 ᫠
qmax 室  襥 ਡ r    樮쭮
஡ p/q,  q<=qmax.

                             10.
     ⭮,   ᫠ A  樮 ⥬ ᫥
 ᭮ p  q   ᪮筮 ਮ᪮ ஡  -
ਮ 2:
                      A = O,(ab)  = O,(ba)                     (*)
                                p         q
 a  b - ࠧ    ⥬ ᫥.
      ணࠬ,    ࠫ ᥫ p
 q (2<=p,q<=30,  p>q) 室  뢮    -
  a  b, 㤮⢮ ᮮ⭮襭 (*). ᫨ ⠪ ,
뢥 ᮮ饭 'ਣ  '.
     ।ᬮ    訡 .
     ਬ砭: 祭 ᫠,   ண    樮
⥬ ᫥  ᭮ S  0,cdef ( c,d,e,f - ),

                       c     d     e     f
                     --- + --- + --- + --- .
                      S^1   S^2   S^3   S^4

                             11.
     । ⢠ K[i] ४७⭮.    K[0]  =  [0,1].
  ᥣ [0,1]    窠 1/3  2/3  㤠 
 ࢠ (1/3,2/3). 稬 ⢮ K[1], 饥  
⠢ ᥣ⮢ [0,1/3]  [2/3,1].     ࠧ 
  (窠 1/9  2/9  ࢮ ᥣ,  窠 7/9 
8/9 -    ண  )    㤠  ।  ࢠ  (1/9,2/9) 
(7/9,8/9).  ࠧ 砥 ⢮ K[2],   ..  
ந ⢮ K[i].   ⠢訩 ᥣ  K[i]
 3   㤠   ᥣ⮢ । ࢠ.  稬,
⠪ ࠧ,  K[i] ⢮ K[i+1].
      3 楫 ᫠ n,a,b.
     室 ।, ਭ  窠  न⮩
a/b  K[n].

                             12.
      訩  騩  ⥫   襥 饥 ⭮
ᥫ a  b.

                             13.
     ᫮ 뢠 ᮢ襭, ᫨  ࠢ 㬬  ᢮
⥫  ᪫祭    ᠬ.    ⭮  ᮢ襭
᫮ ।⠢  
                   p-1    p
                  2   * (2  - 1),   ࠫ쭮 ᫮.
      筮 ।⠢    ᨬ쭮 ᮢ襭
⭮ ᫠ 襣  N.

                             14.
      ࠫ ᫠ E,K,M,T   娬᪮  ॠ樨
                       k + Y -> YmAt + X,
 A,X,Y - ⮬  㯯 ⮬.    ,  
室 ⠪ ࠫ 樥, ⮡  "५" 
뫮   ࠢ⢠.

                             15.
      楫 ᫠ a  b.    㣮쭨 ABC न-
 A=(0,0),  B=(a,b),    न C=(x,y) - 楫 ᫠, 
頤 㣮쭨 ABC  ࠢ .
       頤   㣮쭨 ABC?

                             16.
      N   楫᫥묨 ꥬ V1, ..., Vn ஢,
⮩   ࠭  .        
  ஢ V ஢ .


                             17.
     ᫨ ᫮  e  (᭮ ࠫ쭮 䬠)  -
 n    ᫥ ⮩.    ᯮ짮-
 ᫮ :
          e=1+1/1!+1/2!+...+1/!...

                             18.
     뢥  ࠭ ᫮ 2^n, n<=10000, n - ࠫ쭮.

                             19.
     । ⢮ ७    0,1,2,...,9 
᫥ N^N (N  ⥯ N), N <= 1000.

                            19.1.
     ।,   ।⠢  ࠫ쭮 ᫮ 
 ந  ᫥⥫ ࠫ ᥫ. -
 ᫠   250 ᨬ.  ᫠ - ஡.

                             20.
        ᫠,  ॢ室騥 N.

                             21.
      N.  室 ,   ᪮쪮 㫥 稢
N!=1*2*3*...*N.

                             22.
      室 ணࠬ   ᫠ N  P. ணࠬ  -
   ⠪ ᨬ쭮 ᫮ M,   N!    P 
⥯ M,     P  ⥯ M+1.
     ਬ砭.
1.᫠ N  P ⠪  ,   ᫠  祭 N!.
2.᫠ N  P  ࠫ묨.

                             23.
     ࠫ쭮 ᫮  N>1  ।⠢    㬬 ࠫ
ᥫ ⠪, ⮡ ந  ᫠ 뫮 ᨬ쭮.

                             24.
       ⥫쭮 ⢨⥫쭮 ᫮ R. -
⥫ ⢨⥫ R1,R2,...,Rn,  Ri<4 ,i=1,...,n,  ⠪,
 R=R1*R2*...*Rn=R1+R2+...+Rn

                             25.
      楫  ᫠ (0),(1),...,(5).   ⢮ ୥
ࠢ
              (5)*X^5 + (4)*X^4 + ... + (0) = 0,
᫨ ⭮,    ୨ - 楫 ᫠, A(0)<>0.

                             26.
     뢥   浪  ⠭  몭 ᮪⨬
஡,  祭  0  1,  ⥫   ॢ
15. ᨢ  ⮬   ᫥.

                             27.
      ࠭,  設 ண ᠭ 楫 ᫠.
      室    ॡ,    , ᠭ
    殢 ਡ ,    ᫠,  ᠭ  
㣮  -  1.
      室  筮 ᫮   㤮⢮-
 ᠭ ᫠,  ⮡   ⠪ 室  뫮 -
, ⮡   設  ६ ᠭ ?  -
 ᭮.

                             28.
    a). 
         p(x)=A[n]*x^n+A[n-1]*x^(n-1)+ ...  +A[1]*x+A[0]
 ᢮ 樥⠬ A[n], ... ,A[0].   祭
P  窥 x.
    b). ᫮  k-筮 ⥬  ᢮ ।⠢
                        (A[n], ... ,A[0]),
..  筮 ⥬   祭
         A[n]*k^n+A[n-1]*k^(n-1)+ ...  +A[1]*k+A[0].
       祭.

                             29.

                                   n
      N- ⥯  A(x) = SUM  a[i] * x^i   ᢮
                                  i=0

樥⠬ a[i].  樥 b[i],i=0,...,n*m, m- -
  A(x). ᫠ n,m<=40.

                             30.
     ᫨ 樥 A[1],  A[2],  ..., A[N]
童
            n        n-1
     P(x) =x + A[1]*x   +...+ A[N-1]*x + A[N]
 묨 ⢨⥫묨 ﬨ X[1], X[2], ..., X[N].

                             31.
                 n
      童 SUM a[i]*x^i  ஬ ᢮ 樥⮢
                i=0

a[i], i=0,..,n. 室 ᫨ 樥 b[i] ⠪ -
童, 

    n              n
   SUM b[i]*x^i = SUM a[i]*(x+d)^i
   i=0            i=0

  d.

                             32.
     ᫨  祭  
                     f(x)=ax^4+bx^3+cx^2+dx+e
 x=1, ..., 10000, ᯮ   51000 権 *,+.
