SMO 1997

$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$ .

Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$ . Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$ . Justify your answers.

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$ . Justify your answers.

Let $n \ge 2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$ . Show that $$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$