Difference between revisions of "1971 Canadian MO Problems"
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== Problem 1 == | == Problem 1 == | ||
<math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>OE</math> to cut the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle | <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>OE</math> to cut the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle | ||
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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. | |
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Show that, for all integers <math>n</math>, <math>n^2+2n+12</math> is not a multiple of 121. | Show that, for all integers <math>n</math>, <math>n^2+2n+12</math> is not a multiple of 121. | ||
+ | |||
[[1971 Canadian MO Problems/Problem 6 | Solution]] | [[1971 Canadian MO Problems/Problem 6 | Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | Let <math>n</math> be a five digit number (whose first digit is non-zero) and let <math>m</math> be the four digit number formed from n by removing its middle digit. Determine all <math>n</math> such that <math>n/m</math> is an integer. | ||
[[1971 Canadian MO Problems/Problem 7 | Solution]] | [[1971 Canadian MO Problems/Problem 7 | Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | A regular pentagon is inscribed in a circle of radius <math>r</math>. <math>P</math> is any point inside the pentagon. Perpendiculars are dropped from <math>P</math> to the sides, or the sides produced, of the pentagon. | ||
+ | a) Prove that the sum of the lengths of these perpendiculars is constant. | ||
+ | |||
+ | b) Express this constant in terms of the radius <math>r</math>. | ||
[[1971 Canadian MO Problems/Problem 8 | Solution]] | [[1971 Canadian MO Problems/Problem 8 | Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
+ | Two flag poles of height <math>h</math> and <math>k</math> are situated <math>2a</math> units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal. | ||
[[1971 Canadian MO Problems/Problem 9 | Solution]] | [[1971 Canadian MO Problems/Problem 9 | Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | + | Suppose that <math>n</math> people each know exactly one piece of information, and all <math>n</math> pieces are different. Every time person <math>A</math> phones person <math>B</math>, <math>A</math> tells <math>B</math> everything that <math>A</math> knows, while <math>B</math> tells <math>A</math> nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum. | |
[[1971 Canadian MO Problems/Problem 10 | Solution]] | [[1971 Canadian MO Problems/Problem 10 | Solution]] |
Latest revision as of 07:58, 13 September 2012
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.
Problem 7
Let be a five digit number (whose first digit is non-zero) and let
be the four digit number formed from n by removing its middle digit. Determine all
such that
is an integer.
Problem 8
A regular pentagon is inscribed in a circle of radius .
is any point inside the pentagon. Perpendiculars are dropped from
to the sides, or the sides produced, of the pentagon.
a) Prove that the sum of the lengths of these perpendiculars is constant.
b) Express this constant in terms of the radius .
Problem 9
Two flag poles of height and
are situated
units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
Problem 10
Suppose that people each know exactly one piece of information, and all
pieces are different. Every time person
phones person
,
tells
everything that
knows, while
tells
nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.