Math 289 – Winter 2006 – Problem Set 9
Due March 22, 2006
1.(10) Find a number divisible by 2 and 9 which has exactly
a) 14 divisors, b) 17 divisors.
2.(10) How many zeros are at the end of 2006! = 1 ·2·3·. . . ·2006?
3.(10) Does the polynomial
P(x) = 1 + x
1+x2
2! +. . . +xn
n!
has a multiple roots (i.e. a real root with a multiplicity bigger than one)?
4.(10) Te diagonals of a convex quadrilateral ABCD intersect in O. Show that
|AB|2+|BC|2+|CD|2+|DA|2= 2 |AO|2+|BO|2+|C O|2+|D O|2
exactly if either AC is perpendicular to BD or one of the diagonals is bisected in O.
5.(15) For every n∈N, find the largest integer kfor which
2kdivides b(3 + √11)2n−1c.
6.(15) An infinite chessboard has the shape of the first quadrant. Is it possible to write
a positive integer into each square, such that each row and each column contains each
positive integer exactly once?
7.(20) Let x, y, z be the lengths of the sides of a triangle, and let
f(x, y, z) =
x−y
x+y+y−z
y+z+z−x
z+x
.
Prove that
a) f(x, y, z)<1,b) f(x, y , z)<1
8,c) find the upper bound of f(x, y, z ).
1