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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2018 Canadian Mathematical Olympiad Qualification
2018 Canadian Mathematical Olympiad Qualification
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
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Cycling cards in a row
Let
n
n
n
and
k
k
k
be positive integers with
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
. A set of cards numbered
1
1
1
to
n
n
n
are arranged randomly in a row from left to right. A person alternates between performing the following moves: [*] The leftmost card in the row is moved
k
−
1
k-1
k
−
1
positions to the right while the cards in positions
2
2
2
through
k
k
k
are each moved one place to the left. [*] The rightmost card in the row is moved
k
−
1
k-1
k
−
1
positions to the left while the cards in positions
n
−
k
+
1
n-k+1
n
−
k
+
1
through
n
−
1
n-1
n
−
1
are each moved one place to the right.Determine the probability that after some number of moves the cards end up in order from
1
1
1
to
n
n
n
, left to right.
7
1
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P(n) = rad(n) for infinitely many n
Let
n
n
n
be a positive integer, with prime factorization
n
=
p
1
e
1
p
2
e
2
⋯
p
r
e
r
n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}
n
=
p
1
e
1
p
2
e
2
⋯
p
r
e
r
for distinct primes
p
1
,
…
,
p
r
p_1, \ldots, p_r
p
1
,
…
,
p
r
and
e
i
e_i
e
i
positive integers. Define
r
a
d
(
n
)
=
p
1
p
2
⋯
p
r
,
rad(n) = p_1p_2\cdots p_r,
r
a
d
(
n
)
=
p
1
p
2
⋯
p
r
,
the product of all distinct prime factors of
n
n
n
. Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with rational coefficients such that there exists infinitely many positive integers
n
n
n
with
P
(
n
)
=
r
a
d
(
n
)
P(n) = rad(n)
P
(
n
)
=
r
a
d
(
n
)
.
6
1
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Sum minus product in Z_3^n is zero
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. Determine the number of
n
n
n
-tuples
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2, \ldots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
such that
x
k
∈
{
0
,
1
,
2
}
x_k \in \{0, 1, 2\}
x
k
∈
{
0
,
1
,
2
}
for
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
and
∑
k
=
1
n
x
k
−
∏
k
=
1
n
x
k
\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k
∑
k
=
1
n
x
k
−
∏
k
=
1
n
x
k
is divisible by
3
3
3
.
5
1
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Infinite palindromes divisible by n
A palindrome is a number that remains the same when its digits are reversed. Let
n
n
n
be a product of distinct primes not divisible by
10
10
10
. Prove that infinitely many multiples of
n
n
n
are palindromes.
4
1
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Sides and diagonals have same lengths
Construct a convex polygon such that each of its sides has the same length as one of its diagonals and each diagonal has the same length as one of its sides, or prove that such a polygon does not exist.
3
1
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Obtuse isosceles iff quadratic has distinct roots
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
B
C
AB = BC
A
B
=
BC
. Prove that
△
A
B
C
\triangle ABC
△
A
BC
is an obtuse triangle if and only if the equation
A
x
2
+
B
x
+
C
=
0
Ax^2 + Bx + C = 0
A
x
2
+
B
x
+
C
=
0
has two distinct real roots, where
A
A
A
,
B
B
B
,
C
C
C
, are the angles in radians.
2
1
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Nesting polygons
We call a pair of polygons,
p
p
p
and
q
q
q
, nesting if we can draw one inside the other, possibly after rotation and/or reflection; otherwise we call them non-nesting.Let
p
p
p
and
q
q
q
be polygons. Prove that if we can find a polygon
r
r
r
, which is similar to
q
q
q
, such that
r
r
r
and
p
p
p
are non-nesting if and only if
p
p
p
and
q
q
q
are not similar.
1
1
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A cubic system of two equations
Determine all real solutions to the following system of equations:
{
y
=
4
x
3
+
12
x
2
+
12
x
+
3
x
=
4
y
3
+
12
y
2
+
12
y
+
3.
\begin{cases} y = 4x^3 + 12x^2 + 12x + 3\\ x = 4y^3 + 12y^2 + 12y + 3. \end{cases}
{
y
=
4
x
3
+
12
x
2
+
12
x
+
3
x
=
4
y
3
+
12
y
2
+
12
y
+
3.