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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1991 Canada National Olympiad
1991 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
4
1
Hide problems
Can ten numbers be chosen so that differences be distinct?
Can ten distinct numbers
a
1
,
a
2
,
b
1
,
b
2
,
b
3
,
c
1
,
c
2
,
d
1
,
d
2
,
d
3
a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3
a
1
,
a
2
,
b
1
,
b
2
,
b
3
,
c
1
,
c
2
,
d
1
,
d
2
,
d
3
be chosen from
{
0
,
1
,
2
,
…
,
14
}
\{0, 1, 2, \ldots, 14\}
{
0
,
1
,
2
,
…
,
14
}
, so that the
14
14
14
differences
∣
a
1
−
b
1
∣
|a_1 - b_1|
∣
a
1
−
b
1
∣
,
∣
a
1
−
b
2
∣
|a_1 - b_2|
∣
a
1
−
b
2
∣
,
∣
a
1
−
b
3
∣
|a_1 - b_3|
∣
a
1
−
b
3
∣
,
∣
a
2
−
b
1
∣
|a_2 - b_1|
∣
a
2
−
b
1
∣
,
∣
a
2
−
b
2
∣
|a_2 - b_2|
∣
a
2
−
b
2
∣
,
∣
a
2
−
b
3
∣
|a_2 - b_3|
∣
a
2
−
b
3
∣
,
∣
c
1
−
d
1
∣
|c_1 - d_1|
∣
c
1
−
d
1
∣
,
∣
c
1
−
d
2
∣
|c_1 - d_2|
∣
c
1
−
d
2
∣
,
∣
c
1
−
d
3
∣
|c_1 - d_3|
∣
c
1
−
d
3
∣
,
∣
c
2
−
d
1
∣
|c_2 - d_1|
∣
c
2
−
d
1
∣
,
∣
c
2
−
d
2
∣
|c_2 - d_2|
∣
c
2
−
d
2
∣
,
∣
c
2
−
d
3
∣
|c_2 - d_3|
∣
c
2
−
d
3
∣
,
∣
a
1
−
c
1
∣
|a_1 - c_1|
∣
a
1
−
c
1
∣
, and
∣
a
2
−
c
2
∣
|a_2 - c_2|
∣
a
2
−
c
2
∣
are all distinct?
3
1
Hide problems
Midpoints of chords on a circle
Let
C
C
C
be a circle and
P
P
P
a given point in the plane. Each line through
P
P
P
which intersects
C
C
C
determines a chord of
C
C
C
. Show that the midpoints of these chords lie on a circle.
2
1
Hide problems
digits in base 2
Let
n
n
n
be a fixed positive integer. Find the sum of all positive integers with the property that in base
2
2
2
each has exactly
2
n
2n
2
n
digits, consisting of
n
n
n
1's and
n
n
n
0's. (The first digit cannot be
0
0
0
.)
1
1
Hide problems
Diophantine equation
Show that the equation
x
2
+
y
5
=
z
3
x^2+y^5=z^3
x
2
+
y
5
=
z
3
has infinitely many solutions in integers
x
,
y
,
z
x, y,z
x
,
y
,
z
for which
x
y
z
≠
0
xyz \neq 0
x
yz
=
0
.
5
1
Hide problems
Find number of existing parallelograms
The sides of an equilateral triangle
A
B
C
ABC
A
BC
are divided into
n
n
n
equal parts
(
n
≥
2
)
.
(n \geq 2) .
(
n
≥
2
)
.
For each point on a side, we draw the lines parallel to other sides of the triangle
A
B
C
,
ABC,
A
BC
,
e.g. for
n
=
3
n=3
n
=
3
we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /* # of vertical lines, including AB */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("
A
A
A
",A,W); label("
C
C
C
",C,NE); label("
B
B
B
",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); } [/asy] For each
n
≥
2
,
n \geq 2,
n
≥
2
,
find the number of existing parallelograms.