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National and Regional Contests
Canada Contests
Canada National Olympiad
1984 Canada National Olympiad
1984 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Classic: 7 real numbers
Given any
7
7
7
real numbers, prove that there are two of them
x
,
y
x,y
x
,
y
such that
0
≤
x
−
y
1
+
x
y
≤
1
3
0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}
0
≤
1
+
x
y
x
−
y
≤
3
1
.
4
1
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Acute triangle with unit area
An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least
2
27
4
\frac{2}{\sqrt[4]{27}}
4
27
2
.
3
1
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Digitally Divisible Integers
An integer is digitally divisible if both of the following conditions are fulfilled:
(
a
)
(a)
(
a
)
None of its digits is zero;
(
b
)
(b)
(
b
)
It is divisible by the sum of its digits e.g.
322
322
322
is digitally divisible. Show that there are infinitely many digitally divisible integers.
2
1
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Alice and Bob are in a hardware store
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: [color=#0000FF]Alice: Are you going to cover your keys? [color=#FF0000]Bob: I would like to, but there are only
7
7
7
colours and I have
8
8
8
keys. [color=#0000FF]Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. [color=#FF0000]Bob: You must be careful what you mean by "next to" or "three keys over from" since you can turn the key ring over and the keys are arranged in a circle. [color=#0000FF]Alice: Even so, you don't need
8
8
8
colours. Problem: What is the smallest number of colours needed to distinguish
n
n
n
keys if all the keys are to be covered.
1
1
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1984 Consecutive Squares
Prove that the sum of the squares of
1984
1984
1984
consecutive positive integers cannot be the square of an integer.