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Difference between revisions of "1970 Canadian MO Problems"

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== Problem 1 ==
 
== Problem 1 ==
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Find all number triples <math>(x,y,z)</math> such that when any of these numbers is added to the product of the other two, the result is 2.
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[[1970 Canadian MO Problems/Problem 1 | Solution]]
 
[[1970 Canadian MO Problems/Problem 1 | Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
Given a triangle <math>ABC</math> with angle <math>A</math> obtuse and with altitudes of length <math>h</math> and <math>k</math> as shown in the diagram, prove that <math>a+h\ge b+k</math>. Find under what conditions <math>a+h=b+k</math>.
  
 +
<asy>
 +
draw((0,0)--(5,0)--(16/5,12/5)--cycle,dot);
 +
draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black);
 +
MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE);
 +
MP("E",(2.5,0),NE);MP("A",(2.5,7.5/4),N);
 +
MP("h",(2.5,7.5/8),W);MP("k",(41/10,6/5),NE);
 +
draw((-.2,.2)--(2.5-.2,7.5/4+.2),arrow=ArcArrow(TeXHead));
 +
draw((2.5-.2,7.5/4+.2)--(-.2,.2),arrow=ArcArrow(TeXHead));
 +
MP("b",(2.3/2-.05,7.5/8+.25),N);
 +
draw((0,-.2)--(5,-.2),arrow=ArcArrow(TeXHead));
 +
draw((5,-.2)--(0,-.2),arrow=ArcArrow(TeXHead));
 +
MP("a",(2.5,-.2),S);
 +
draw((16/5,12/5)--(16/5-.2,12/5-.15)--(16/5-.2+.15,12/5-.15-.2)--(16/5+.15,12/5-.2)--cycle,black);
 +
</asy>
  
 +
[[1970 Canadian MO Problems/Problem 2 | Solution]]
  
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[[1970 Canadian MO Problems/Problem 2 | Solution]]
 
 
== Problem 3 ==
 
== Problem 3 ==
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A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.
−
 
 
−
 
 
  
 
[[1970 Canadian MO Problems/Problem 3 | Solution]]
 
[[1970 Canadian MO Problems/Problem 3 | Solution]]
 
== Problem 4 ==
 
== Problem 4 ==
 +
a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is <math>1/25</math> of the original integer.
  
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+
b) Show that there is no integer such that the deletion of the first digit produces a result that is <math>1/35</math> of the original integer.
  
 
[[1970 Canadian MO Problems/Problem 4 | Solution]]
 
[[1970 Canadian MO Problems/Problem 4 | Solution]]
 
== Problem 5 ==
 
== Problem 5 ==
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−
 
 
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> of the sides of the quadrilateral satisfy the inequalities <math>2\le a^2+b^2+c^2+d^2\le 4.</math>
 
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> of the sides of the quadrilateral satisfy the inequalities <math>2\le a^2+b^2+c^2+d^2\le 4.</math>
  
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== Problem 6 ==
 
== Problem 6 ==
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Given three non-collinear points <math>A,B,C</math>, construct a circle with centre <math>C</math> such that the tangents from <math>A</math> and <math>B</math> are parallel.
  
 
[[1970 Canadian MO Problems/Problem 6 | Solution]]
 
[[1970 Canadian MO Problems/Problem 6 | Solution]]
 
== Problem 7 ==
 
== Problem 7 ==
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+
Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
  
 
[[1970 Canadian MO Problems/Problem 7 | Solution]]
 
[[1970 Canadian MO Problems/Problem 7 | Solution]]
 
== Problem 8 ==
 
== Problem 8 ==
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+
Consider all line segments of length 4 with one end-point on the line <math>y=x</math> and the other end-point on the line <math>y=2x</math>. Find the equation of the locus of the midpoints of these line segments.
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[[1970 Canadian MO Problems/Problem 8 | Solution]]
 
[[1970 Canadian MO Problems/Problem 8 | Solution]]
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== Problem 9 ==
 
== Problem 9 ==
 +
Let <math>f(n)</math> be the sum of the first <math>n</math> terms of the sequence
 +
<cmath> 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, .  </cmath>
 +
a) Give a formula for <math>f(n)</math>.
  
 +
b) Prove that <math>f(s+t)-f(s-t)=st</math> where <math>s</math> and <math>t</math> are positive integers and <math>s>t</math>.
  
 
[[1970 Canadian MO Problems/Problem 9 | Solution]]
 
[[1970 Canadian MO Problems/Problem 9 | Solution]]
 
== Problem 10 ==
 
== Problem 10 ==
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Given the polynomial
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<cmath> f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n </cmath>
 +
with integer coefficients <math>a_1,a_2,\ldots,a_n</math>, and given also that there exist four distinct integers <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> such that
 +
<cmath>f(a)=f(b)=f(c)=f(d)=5, </cmath>
 +
show that there is no integer <math>k</math> such that <math>f(k)=8</math>.
  
 
[[1970 Canadian MO Problems/Problem 10 | Solution]]
 
[[1970 Canadian MO Problems/Problem 10 | Solution]]
−
== Resources ==
+
== See Also ==
  
 
* [[1970 Canadian MO]]
 
* [[1970 Canadian MO]]
 
* [[Canadian Mathematical Olympiad]]
 
* [[Canadian Mathematical Olympiad]]
 
* [[Canadian MO Problems and Solutions]]
 
* [[Canadian MO Problems and Solutions]]

Latest revision as of 03:32, 8 October 2014

Problem 1

Find all number triples $(x,y,z)$ such that when any of these numbers is added to the product of the other two, the result is 2.

Solution

Problem 2

Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.

[asy] draw((0,0)--(5,0)--(16/5,12/5)--cycle,dot); draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black); MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE); MP("E",(2.5,0),NE);MP("A",(2.5,7.5/4),N); MP("h",(2.5,7.5/8),W);MP("k",(41/10,6/5),NE); draw((-.2,.2)--(2.5-.2,7.5/4+.2),arrow=ArcArrow(TeXHead)); draw((2.5-.2,7.5/4+.2)--(-.2,.2),arrow=ArcArrow(TeXHead)); MP("b",(2.3/2-.05,7.5/8+.25),N); draw((0,-.2)--(5,-.2),arrow=ArcArrow(TeXHead)); draw((5,-.2)--(0,-.2),arrow=ArcArrow(TeXHead)); MP("a",(2.5,-.2),S); draw((16/5,12/5)--(16/5-.2,12/5-.15)--(16/5-.2+.15,12/5-.15-.2)--(16/5+.15,12/5-.2)--cycle,black); [/asy]

Solution

Problem 3

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

Solution

Problem 4

a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is $1/25$ of the original integer.

b) Show that there is no integer such that the deletion of the first digit produces a result that is $1/35$ of the original integer.

Solution

Problem 5

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities $2\le a^2+b^2+c^2+d^2\le 4.$

Solution

Problem 6

Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.

Solution

Problem 7

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

Solution

Problem 8

Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.

Solution

Problem 9

Let $f(n)$ be the sum of the first $n$ terms of the sequence \[0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, .\] a) Give a formula for $f(n)$.

b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.

Solution

Problem 10

Given the polynomial \[f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n\] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[f(a)=f(b)=f(c)=f(d)=5,\] show that there is no integer $k$ such that $f(k)=8$.

Solution

See Also

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