2011 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2011 Problem 6

\Gamma

x> 0

of the hyperbola

x^2 - y^2 = 1.

P_0, P_1,..., P_n

different points of

\Gamma

P_0 = (1, 0)

P_1 = (13/12, 5/12)

t_i

be the tangent line to

\Gamma

P_i

. Suppose that for all

i \geq 0

the area of the region bounded by

t_i, t_{i +1}

\Gamma

is a constant independent of

i

. Find the coordinates of the points

P_i

1

CIIM 2011 Problem 5

n

be a positive integer with

d

digits, all different from zero. For

k = 0,. . . , d - 1

n_k

as the number obtained by moving the last

k

n

to the beginning. For example, if

n = 2184

n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842

m

a positive integer, define

s_m(n)

as the number of values

k

n_k

is a multiple of

m.

Finally, define

a_d

as the number of integers

n

d

digits all nonzero, for which

s_2 (n) + s_3 (n) + s_5 (n) = 2d.

\lim_{d \to \infty} \frac{a_d}{5^d}.

1

CIIM 2011 Problem 4

n \geq 3

(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).

C_n = (c_{i, j})

n \times n

matrix defined by

c_{i, j} = b _{(j -i) \mod n}

\det (C_n) = 3

n

is not a multiple of 3 and

\det (C_n) = 0

n

is a multiple of 3.

1

CIIM 2011 Problem 3

f(x)

be a rational function with complex coefficients whose denominator does not have multiple roots. Let

u_0, u_1,... , u_n

be the complex roots of

f

w_1, w_2,..., w_m

be the roots of

f'

u_0

is a simple root of

f

\sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.

1

CIIM 2011 Problem 2

k

be a positive integer, and let

a

be an integer such that

a-2

is a multiple of

7

a^6-1

is a multiple of

7^k

(a + 1)^6-1

is also a multiple of

7^k

1

CIIM 2011 Problem 1

Find all real numbers

a

for which there exist different real numbers

b, c, d

a

such that the four tangents drawn to the curve

y = \sin (x)

(a, \sin (a)), (b, \sin (b)), (c, \sin (c))

(d, \sin (d))

form a rectangle.