CIIM 2011

Find all real numbers $a$ for which there exist different real numbers $b, c, d$ different from $a$ such that the four tangents drawn to the curve $y = \sin (x)$ at the points $(a, \sin (a)), (b, \sin (b)), (c, \sin (c))$ and $(d, \sin (d))$ form a rectangle.

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$ .

Prove that $(a + 1)^6-1$ is also a multiple of $7^k$ .

Let $f(x)$ 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 $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$ . Suppose that $u_0$ is a simple root of $f$ . Prove that $$\sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.$$

For $n \geq 3$ , let $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$ Let $C_n = (c_{i, j})$ the $n \times n$ matrix defined by $c_{i, j} = b _{(j -i) \mod n}$ . Show that $\det (C_n) = 3$ if $n$ is not a multiple of 3 and $\det (C_n) = 0$ if $n$ is a multiple of 3.

Let $n$ be a positive integer with $d$ digits, all different from zero. For $k = 0,. . . , d - 1$ , we define $n_k$ as the number obtained by moving the last $k$ digits of $n$ to the beginning. For example, if $n = 2184$ then $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$ . For $m$ a positive integer, define $s_m(n)$ as the number of values $k$ such that $n_k$ is a multiple of $m.$ Finally, define $a_d$ as the number of integers $n$ with $d$ digits all nonzero, for which $s_2 (n) + s_3 (n) + s_5 (n) = 2d.$ Find $$\lim_{d \to \infty} \frac{a_d}{5^d}.$$

Let $\Gamma$ be the branch $x> 0$ of the hyperbola $x^2 - y^2 = 1.$ Let $P_0, P_1,..., P_n$ different points of $\Gamma$ with $P_0 = (1, 0)$ and $P_1 = (13/12, 5/12)$ . Let $t_i$ be the tangent line to $\Gamma$ at $P_i$ . Suppose that for all $i \geq 0$ the area of the region bounded by $t_i, t_{i +1}$ and $\Gamma$ is a constant independent of $i$ . Find the coordinates of the points $P_i$ .