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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2022 Canadian Mathematical Olympiad Qualification
2022 Canadian Mathematical Olympiad Qualification
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
4
1
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exist infinitely many non-constant polynomials that are not n-good for any n
For a non-negative integer
n
n
n
, call a one-variable polynomial
F
F
F
with integer coefficients
n
n
n
-good if: (a)
F
(
0
)
=
1
F(0) = 1
F
(
0
)
=
1
(b) For every positive integer
c
c
c
,
F
(
c
)
>
0
F(c) > 0
F
(
c
)
>
0
, and (c) There exist exactly
n
n
n
values of
c
c
c
such that
F
(
c
)
F(c)
F
(
c
)
is prime. Show that there exist infinitely many non-constant polynomials that are not
n
n
n
-good for any
n
n
n
.
3
1
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\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) >= 0
Consider n real numbers
x
0
,
x
1
,
.
.
.
,
x
n
−
1
x_0, x_1, . . . , x_{n-1}
x
0
,
x
1
,
...
,
x
n
−
1
for an integer
n
≥
2
n \ge 2
n
≥
2
. Moreover, suppose that for any integer
i
i
i
,
x
i
+
n
=
x
i
x_{i+n} = x_i
x
i
+
n
=
x
i
. Prove that
∑
i
=
0
n
−
1
x
i
(
3
x
i
−
4
x
i
+
1
+
x
i
+
2
)
≥
0.
\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) \ge 0.
i
=
0
∑
n
−
1
x
i
(
3
x
i
−
4
x
i
+
1
+
x
i
+
2
)
≥
0.
2
1
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m^2 + n and n^2 + m are both perfect squares
Determine all pairs of integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
m
2
+
n
m^2 + n
m
2
+
n
and
n
2
+
m
n^2 + m
n
2
+
m
are both perfect squares.
5
1
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Alice has 4 boxes, 327 blue balls, and 2022 red balls
Alice has four boxes,
327
327
327
blue balls, and
2022
2022
2022
red balls. The blue balls are labeled
1
1
1
to
327
327
327
. Alice first puts each of the balls into a box, possibly leaving some boxes empty. Then, a random label between
1
1
1
and
327
327
327
(inclusive) is selected, Alice finds the box the ball with the label is in, and selects a random ball from that box. What is the maximum probability that she selects a red ball?
1
1
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finally... amogus
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. On a spaceship, there are
n
n
n
crewmates. At most one accusation of being an imposter can occur from one crewmate to another crewmate. Multiple accusations are thrown, with the following properties: • Each crewmate made a different number of accusations. • Each crewmate received a different number of accusations. • A crewmate does not accuse themself. Prove that no two crewmates made accusations at each other.
7
1
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Chase your Dreams
Let
A
B
C
ABC
A
BC
be a triangle with
∣
A
B
∣
<
∣
A
C
∣
|AB| < |AC|
∣
A
B
∣
<
∣
A
C
∣
, where
∣
⋅
∣
| · |
∣
⋅
∣
denotes length. Suppose
D
,
E
,
F
D, E, F
D
,
E
,
F
are points on side
B
C
BC
BC
such that
D
D
D
is the foot of the perpendicular on
B
C
BC
BC
from
A
A
A
,
A
E
AE
A
E
is the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
, and
F
F
F
is the midpoint of
B
C
BC
BC
. Further suppose that
∠
B
A
D
=
∠
D
A
E
=
∠
E
A
F
=
∠
F
A
C
\angle BAD = \angle DAE = \angle EAF = \angle FAC
∠
B
A
D
=
∠
D
A
E
=
∠
E
A
F
=
∠
F
A
C
. Determine all possible values of
∠
A
B
C
\angle ABC
∠
A
BC
.
8
1
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Coining a Grid
Let
{
m
,
n
,
k
}
\{m, n, k\}
{
m
,
n
,
k
}
be positive integers.
{
k
}
\{k\}
{
k
}
coins are placed in the squares of an
m
×
n
m \times n
m
×
n
grid. A square may contain any number of coins, including zero. Label the
{
k
}
\{k\}
{
k
}
coins
C
1
,
C
2
,
⋅
⋅
⋅
C
k
C_1, C_2, · · · C_k
C
1
,
C
2
,⋅⋅⋅
C
k
. Let
r
i
r_i
r
i
be the number of coins in the same row as
C
i
C_i
C
i
, including
C
i
C_i
C
i
itself. Let
s
i
s_i
s
i
be the number of coins in the same column as
C
i
C_i
C
i
, including
C
i
C_i
C
i
itself. Prove that
∑
i
=
1
k
1
r
i
+
s
i
≤
m
+
n
4
\sum_{i=1}^k \frac{1}{r_i+s_i} \leq \frac{m+n}{4}
i
=
1
∑
k
r
i
+
s
i
1
≤
4
m
+
n
6
1
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a+b+c=1/a+1/b+1/c=3, prove that at least one of a,b,c is negative
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers, which are not all equal, such that
a
+
b
+
c
=
1
a
+
1
b
+
1
c
=
3.
a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.
a
+
b
+
c
=
a
1
+
b
1
+
c
1
=
3.
Prove that at least one of
a
,
b
,
c
a, b, c
a
,
b
,
c
is negative.