Canadian Mathematical Olympiad
1995
PROBLEM 1
Letf(x) = 9
x
9 x
+3. Evaluate the sum f( 1
1996) +f( 2
1996) +f( 3
1996) ++f(1995
1996)
PROBLEM 2
Leta,b, andcbe positive real numbers. Prove that
a a
b b
c c
(abc) a+b+c
3
:
PROBLEM 3
Dene a boomerang as a quadrilateral whose op-posite sides do not intersect and one of whose in-ternal angles is greater than 180 degrees. (See Figure displayed.) Let C be a convex polygon
having 5 sides. Suppose that the interior region of C is the union ofq quadrilaterals, none of whose
interiors intersect one another. Also suppose that
b of these quadrilaterals are boomerangs. Show
thatqb+ s,2
2 . PROBLEM 4
Let n be a xed positive integer. Show that for only nonnegative integersk, the
diophantine equation
x 3
1+ x
3
2+ +x
3
n= y
3k +2
has innitely many solutions in positive integersx i and
y. PROBLEM 5
Suppose thatuis a real parameter with 0<u<1. Dene
f(x) = (
0 if 0xu
1,
p
ux+ p
(1,u)(1,x)
2
ifux1
and dene the sequencefu n
grecursively as follows: u
1=
f(1); andu n=
f(u
n,1) for all n>1:
Show that there exists a positive inegerkfor whichu k= 0.
