Difference between revisions of "1969 Canadian MO Problems"
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== Problem 9 == | == Problem 9 == | ||
Latest revision as of 12:46, 8 October 2007
Contents
Problem 1
Show that if and
are not all zero, then
for every positive integer
Problem 2
Determine which of the two numbers ,
is greater for any
.
Problem 3
Let be the length of the hypotenuse of a right triangle whose two other sides have lengths
and
. Prove that
. When does the equality hold?
Problem 4
Let be an equilateral triangle, and
be an arbitrary point within the triangle. Perpendiculars
are drawn to the three sides of the triangle. Show that, no matter where
is chosen,
.
Problem 5
Let be a triangle with sides of length
,
and
. Let the bisector of the
cut
at
. Prove that the length of
is
Problem 6
Find the sum of , where
.
Problem 7
Show that there are no integers for which
.
Problem 8
Let be a function with the following properties:
1) is defined for every positive integer
;
2) is an integer;
3) ;
4) for all
and
;
5) whenever
.
Prove that .
Problem 9
Show that for any quadrilateral inscribed in a circle of radius the length of the shortest side is less than or equal to
.
Problem 10
Let be the right-angled isosceles triangle whose equal sides have length 1.
is a point on the hypotenuse, and the feet of the perpendiculars from
to the other sides are
and
. Consider the areas of the triangles
and
, and the area of the rectangle
. Prove that regardless of how
is chosen, the largest of these three areas is at least
.