AT-OEMO-REG 2000

For which natural numbers $n$ does $2^n > 10n^2 -60n + 80$ hold?

For any real number $a$ , find all real numbers $x$ that satisfy the following equation. $$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$

We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$ ). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$ ?

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$ . (a) Determine the terms of the sequence for $u_1 = 1$ . (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.