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 2 == | == Problem 2 == | ||
| − | + | Let <math>x</math> and <math>y</math> be positive real numbers such that <math>x+y=1</math>. Show that <math>\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9</math>. | |
[[1971 Canadian MO Problems/Problem 2 | Solution]] | [[1971 Canadian MO Problems/Problem 2 | Solution]] | ||
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| − | + | <math>ABCD</math> is a quadrilateral with <math>AD=BC</math>. If <math>\angle ADC</math> is greater than <math>\angle BCD</math>, prove that <math>AC>BD</math>. | |
[[1971 Canadian MO Problems/Problem 3 | Solution]] | [[1971 Canadian MO Problems/Problem 3 | Solution]] | ||
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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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== Problem 6 == | == Problem 6 == | ||
| + | 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]] | ||
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== 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. | ||
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| + | 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
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.
