OBM 2001

Show that for any $a,b,c$ positive reals,

$$(a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)}$$

Given $a_0 > 1$ , the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$ , $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers.

$ABC$ is a triangle $E, F$ are points in $AB$ , such that $AE = EF = FB$

$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$ $AD$ is perpendicular to $CF$ . The angle CFA is the triple of angle BDF. ( $3\angle BDF = \angle CFA$ )

Determine the ratio $\frac{DB}{DC}$ .

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A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.

A one-player game is played as follows: There is a bowl at each integer on the $Ox$ -axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either

(A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or

(B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left.

Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$ ).