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Problems
Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2014 Canadian Mathematical Olympiad Qualification
2014 Canadian Mathematical Olympiad Qualification
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
Hide problems
An integer with more than n! divisors
For any given non-negative integer
m
m
m
, let
f
(
m
)
f(m)
f
(
m
)
be the number of
1
1
1
's in the base
2
2
2
representation of
m
m
m
. Let
n
n
n
be a positive integer. Prove that the integer
∑
m
=
0
2
n
−
1
(
(
−
1
)
f
(
m
)
⋅
2
m
)
\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)
m
=
0
∑
2
n
−
1
(
(
−
1
)
f
(
m
)
⋅
2
m
)
contains at least
n
!
n!
n
!
positive divisors.
7
1
Hide problems
Six bugs walking
A bug is standing at each of the vertices of a regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
. At the same time each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in originally to the vertex it picked. All bugs travel at the same speed and are of negligible size. Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move to the vertices so that no two bugs are ever in the same spot at the same time?
6
1
Hide problems
Midpoints and isosceles triangles
Given a triangle
A
,
B
,
C
,
X
A, B, C, X
A
,
B
,
C
,
X
is on side
A
B
AB
A
B
,
Y
Y
Y
is on side
A
C
AC
A
C
, and
P
P
P
and
Q
Q
Q
are on side
B
C
BC
BC
such that
A
X
=
A
Y
,
B
X
=
B
P
AX = AY , BX = BP
A
X
=
A
Y
,
BX
=
BP
and
C
Y
=
C
Q
CY = CQ
C
Y
=
CQ
. Let
X
P
XP
XP
and
Y
Q
YQ
Y
Q
intersect at
T
T
T
. Prove that
A
T
AT
A
T
passes through the midpoint of
P
Q
PQ
PQ
.
5
1
Hide problems
Roots of an irreducible quartic
Let
f
(
x
)
=
x
4
+
2
x
3
−
x
−
1
f(x) = x^4 + 2x^3 - x - 1
f
(
x
)
=
x
4
+
2
x
3
−
x
−
1
.(a) Prove that
f
(
x
)
f(x)
f
(
x
)
cannot be written as the product of two non-constant polynomials with integer coefficients.(b) Find the exact values of the 4 roots of
f
(
x
)
f(x)
f
(
x
)
.
4
1
Hide problems
Triangle geometry with lasers
In
△
A
B
C
\triangle ABC
△
A
BC
, the interior sides of which are mirrors, a laser is placed at point
A
1
A_1
A
1
on side
B
C
BC
BC
. A laser beam exits the point
A
1
A_1
A
1
, hits side
A
C
AC
A
C
at point
B
1
B_1
B
1
, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side
A
B
AB
A
B
at point
C
1
C_1
C
1
, then side
B
C
BC
BC
at point
A
2
A_2
A
2
, then side
A
C
AC
A
C
again at point
B
2
B_2
B
2
, then side
A
B
AB
A
B
again at point
C
2
C_2
C
2
, then side
B
C
BC
BC
again at point
A
3
A_3
A
3
, and finally, side
A
C
AC
A
C
again at point
B
3
B_3
B
3
.(a) Prove that
∠
B
3
A
3
C
=
∠
B
1
A
1
C
\angle B_3A_3C = \angle B_1A_1C
∠
B
3
A
3
C
=
∠
B
1
A
1
C
.(b) Prove that such a laser exists if and only if all the angles in
△
A
B
C
\triangle ABC
△
A
BC
are less than
9
0
∘
90^{\circ}
9
0
∘
.
3
1
Hide problems
1111 divides four-digit number plus product of digits
Let
1000
≤
n
=
ABCD
10
≤
9999
1000 \leq n = \text{ABCD}_{10} \leq 9999
1000
≤
n
=
ABCD
10
≤
9999
be a positive integer whose digits
ABCD
\text{ABCD}
ABCD
satisfy the divisibility condition:
1111
∣
(
ABCD
+
AB
×
CD
)
.
1111 | (\text{ABCD} + \text{AB} \times \text{CD}).
1111∣
(
ABCD
+
AB
×
CD
)
.
Determine the smallest possible value of
n
n
n
.
2
1
Hide problems
Keys in safes
Alphonse and Beryl play a game involving
n
n
n
safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the
n
n
n
keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects
m
m
m
of the safes (where
m
<
n
m < n
m
<
n
), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these
m
m
m
safes and tries to use these keys to open up the other
n
−
m
n - m
n
−
m
safes. If he can open a safe with one of the
m
m
m
keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let
P
m
(
n
)
P_m(n)
P
m
(
n
)
be the probability that Alphonse can eventually open all
n
n
n
safes starting from his initial selection of
m
m
m
keys.(a) Show that
P
2
(
3
)
=
2
3
P_2(3) = \frac23
P
2
(
3
)
=
3
2
.(b) Prove that
P
1
(
n
)
=
1
n
P_1(n) = \frac1n
P
1
(
n
)
=
n
1
.(c) For all integers
n
≥
2
n \geq 2
n
≥
2
, prove that
P
2
(
n
)
=
2
n
⋅
P
1
(
n
−
1
)
+
n
−
2
n
⋅
P
2
(
n
−
1
)
.
P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).
P
2
(
n
)
=
n
2
⋅
P
1
(
n
−
1
)
+
n
n
−
2
⋅
P
2
(
n
−
1
)
.
(d) Determine a formula for
P
2
(
n
)
P_2 (n)
P
2
(
n
)
.
1
1
Hide problems
GCD of two functions a constant polynomial
Let
f
:
Z
→
Z
+
f : \mathbb{Z} \rightarrow \mathbb{Z}^+
f
:
Z
→
Z
+
be a function, and define
h
:
Z
×
Z
→
Z
+
h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+
h
:
Z
×
Z
→
Z
+
by
h
(
x
,
y
)
=
gcd
(
f
(
x
)
,
f
(
y
)
)
h(x, y) = \gcd (f(x), f(y))
h
(
x
,
y
)
=
g
cd
(
f
(
x
)
,
f
(
y
))
. If
h
(
x
,
y
)
h(x, y)
h
(
x
,
y
)
is a two-variable polynomial in
x
x
x
and
y
y
y
, prove that it must be constant.