Difference between revisions of "1971 Canadian MO Problems"
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== Problem 5 == | == Problem 5 == | ||
− | + | Let <math>p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0</math>, where the coefficients <math> a_i</math> are integers. If <math>p(0)</math> and <math>p(1)</math> are both odd, show that <math>p(x)</math> has no integral roots. | |
Revision as of 21:49, 13 December 2011
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Contents
[hide]Problem 1
is a chord of a circle such that
and
Let
be the center of the circle. Join
and extend
to cut the circle at
Given
find the radius of the circle
Problem 2
Let and
be positive real numbers such that
. Show that
.
Problem 3
is a quadrilateral with
. If
is greater than
, prove that
.
Problem 4
Determine all real numbers such that the two polynomials
and
have at least one root in common.
Problem 5
Let , where the coefficients
are integers. If
and
are both odd, show that
has no integral roots.
Problem 6
Show that, for all integers ,
is not a multiple of 121.
Solution