IMC2014 Day2 Solutions

8/9/2019 IMC2014 Day2 Solutions

1/6

x n dn (x) dn (x)

{0, 1, . . . , 9} x =

n =1dn (x)10n 1 an

n =1

dn (an )

n =1

(an )n =1

n0 dn (an ) = 1 n n0

an +1 , a n +2 , . . . > 10n n n0. (1)

an =

k =1dk (an )10k 1 dn (an )10n 1 dn (an ) 1

an 10n 1 m > n am 10m > 10n (1) n a1, a2, . . . , a n

[1, 10n ] 10n n lim(10n n) =

a1, a2, . . .

(an ) dn (an ) = 0 n > n 0

gn =2 dn (xn ) = 11 dn (xn ) = 1 .

gn = dn (an ) n

xk =k

n =1

gn 10n 1 k = 1, 2, . . . .

xk +1 10k > x k (xk ) xn 0 , xn 0 +1 , xn 0 +2 , . . .

(an ) xk = an n 1 k n0 k n dn (xk ) = gn = dn (an ) xk = an

n > k n0 dn (xk ) = 0 = dn (an ) xk = an

A = ( a ij )ni,j =1 n n 1, 2, . . . , n

1 i


2/6

A

n

i=1

a ii =n

i=1

i ,

n

i=1

a2ii + 2i 0. g (x) < 0 0 < x < 2 x > 0


3/6

sin xx

(n )

= dn

dxn 1

0 cos(xt )dt =

1

0

n

x n ( cos(xt )) d t =

1

0tn gn (xt )dt

gn (u)

sin u

cos u

n

|gn | 1

sin xx

(n )

1

0tn gn (xt ) dt <

1

0tn dt =

1n + 1

.

R n k

k

k

k

k n d(k, n) k R n k d(k, n)

d(k, n) =k n k, n > 1k + n

n = 1 k = 1 R n

n + 1

n + 1 n + 1

k,n > 1 k n k k

k

n(k 1) n(k 1) k (n 1)(k 1) 1 n, k > 1

i xi = 0 k 1 k d(k, n) kn

d(k, n) kn k,n > 1 k k kn

k k R n kn

k k


4/6

k nk n

nk + 1 O

1

nk 1 k+1 O

k n O k

1, 2,...,n k O O k n

O k

n Dn (x1, . . . , x n ) (1, 2, . . . , n ) x j = j 1 j n 1 k n2

( n, k ) (x1, . . . , x n ) (1, 2, . . . , n ) xi = k + i 1 i k x j = j 1 j n

( n, k ) =k 1

i=0

k 1i

D (n +1) (k + i )n (k + i)

.

a r {i1, . . . , i k } {a1, . . . , a k } ar = is s = r

as {i1, . . . , i k } as = it x = ( x1, . . . , x n ) xi j = a j x = ( x 1, . . . , x i t 1, x

i t +1 , x

n )

[n] \ { i t } xi j = a j j = t a j = a j j = s

a s = at x = ( x1, . . . , x n ) [n] xi j = a j {i1, . . . , i k } {a1, . . . , a k } x = ( x 1, . . . , x i t 1, x

i t +1 , x

n )

[n] \ { i t } xi j = a j {i1, . . . , i k } \ { i t } {a 1, . . . , a k } \ { a t }

1 a s / {i1, . . . , i k } x = ( x1, . . . , x n ) xi j = a j x = ( x 1, . . . , x a s 1, x

a s +1 , x

n )


5/6

[n] \ {as } xi j = a j j = s x = ( x1, . . . , x n ) [n] xi j = a j

{i1, . . . , i k } {a1, . . . , a k } x = ( x 1, . . . , x

a s 1, x

a s +1 , x

n ) [n] \ { a s } xi j = a j

{i1, . . . , i k } \ { is } {a1, . . . , a k } \ { a s } 1 ( n,k,) = ( n 1, k 1, 1)

( n,k,) = ( n , k ,0).

= 0 ( n,k, 0) 2k n k = 0 ( n, 0, 0) = Dn k 1

( n,k, 0) = ( n 1, k 1, 0) + ( n 2, k 1, 0).

x = ( x1, . . . , x n ) xi j = a j xa 1 = i1 xa 1 = i1

[n] \ { i1, a1} {i2, . . . , i k } {a2, . . . , a k } 0 ( n 2, k 1, 0)

[n] \ { a1} {i2, . . . , i k } {a2, . . . , a k } 0 ( n 1, k 1, 0)

k

( n,k, 0) =k 1

i=0

k 1i

D (n +1) (k + i )n (k + i)

, 2 2k n.

k = 1

( n, 1, 0) = ( n 1, 0, 0) + ( n 2, 0, 0) = Dn 1 + Dn 2 = Dnn 1

.


6/6

k 1

( n,k, 0) = ( n 1, k 1, 0) + ( n 2, k 1, 0)

=k 2

i=0

k 2

i

Dn (k 1+ i )

(n 1) (k 1 + i) +

k 2

i=0

k 2

i

D(n 1) (k 1+ i )

(n 2) (k 1 + i)

=k 2

i=0

k 2i

D (n +1) (k + i )n (k + i)

+k 1

i=1

k 2i 1

Dn (k + i 1)(n 1) (k + i 1)

= D(n +1) k

n k +

k 2

i=1

k 2i

D (n +1) (k + i )n (k + i)

+D (n +1) (2k 1)n (2k 1)

+k 2

i=1

k 2i 1

D (n +1) (k + i )n (k + i)

= D(n +1) k

n k +

k 2

i=1

k 2i +

k 2i 1

D (n +1) (k + i )n (k + i) +

D (n +1) (2k 1)n (2k 1)

= D(n +1) k

n k +

k 2

i=1

k 1i

D (n +1) (k + i )n (k + i)

+ D(n +1) (2k 1)n (2k 1)

=k 1

i=0

k 1i

D (n +1) (k + i )n (k + i)

.

n = 2ki j = j a j = k + j j = 1 , . . . , k x = ( x 1 , . . . , x n )

x i j = a j x = ( x k +1 , . . . , x n ) [k] x k !

k 1i=0

k 1i

D k +1 ik i = k!

Documents

IMC2014 Day2 Solutions