MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2013 Canada National Olympiad
2013 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
4
1
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Minimum and Maximum Inequality
Let
n
n
n
be a positive integer. For any positive integer
j
j
j
and positive real number
r
r
r
, define
f
j
(
r
)
f_j(r)
f
j
(
r
)
and
g
j
(
r
)
g_j(r)
g
j
(
r
)
by
f
j
(
r
)
=
min
(
j
r
,
n
)
+
min
(
j
r
,
n
)
,
and
g
j
(
r
)
=
min
(
⌈
j
r
⌉
,
n
)
+
min
(
⌈
j
r
⌉
,
n
)
,
f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),
f
j
(
r
)
=
min
(
j
r
,
n
)
+
min
(
r
j
,
n
)
,
and
g
j
(
r
)
=
min
(⌈
j
r
⌉
,
n
)
+
min
(
⌈
r
j
⌉
,
n
)
,
where
⌈
x
⌉
\lceil x\rceil
⌈
x
⌉
denotes the smallest integer greater than or equal to
x
x
x
. Prove that
∑
j
=
1
n
f
j
(
r
)
≤
n
2
+
n
≤
∑
j
=
1
n
g
j
(
r
)
\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)
j
=
1
∑
n
f
j
(
r
)
≤
n
2
+
n
≤
j
=
1
∑
n
g
j
(
r
)
for all positive real numbers
r
r
r
.
5
1
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Circumcircles and Reflections
Let
O
O
O
denote the circumcentre of an acute-angled triangle
A
B
C
ABC
A
BC
. Let point
P
P
P
on side
A
B
AB
A
B
be such that
∠
B
O
P
=
∠
A
B
C
\angle BOP = \angle ABC
∠
BOP
=
∠
A
BC
, and let point
Q
Q
Q
on side
A
C
AC
A
C
be such that
∠
C
O
Q
=
∠
A
C
B
\angle COQ = \angle ACB
∠
COQ
=
∠
A
CB
. Prove that the reflection of
B
C
BC
BC
in the line
P
Q
PQ
PQ
is tangent to the circumcircle of triangle
A
P
Q
APQ
A
PQ
.
3
1
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Intersections of Circumcircles and Hypotenuse
Let
G
G
G
be the centroid of a right-angled triangle
A
B
C
ABC
A
BC
with
∠
B
C
A
=
9
0
∘
\angle BCA = 90^\circ
∠
BC
A
=
9
0
∘
. Let
P
P
P
be the point on ray
A
G
AG
A
G
such that
∠
C
P
A
=
∠
C
A
B
\angle CPA = \angle CAB
∠
CP
A
=
∠
C
A
B
, and let
Q
Q
Q
be the point on ray
B
G
BG
BG
such that
∠
C
Q
B
=
∠
A
B
C
\angle CQB = \angle ABC
∠
CQB
=
∠
A
BC
. Prove that the circumcircles of triangles
A
Q
G
AQG
A
QG
and
B
P
G
BPG
BPG
meet at a point on side
A
B
AB
A
B
.
2
1
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Partial Sums of Sequence
The sequence
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
consists of the numbers
1
,
2
,
…
,
n
1, 2, \dots, n
1
,
2
,
…
,
n
in some order. For which positive integers
n
n
n
is it possible that the
n
+
1
n+1
n
+
1
numbers
0
,
a
1
,
a
1
+
a
2
,
a
1
+
a
2
+
a
3
,
…
,
a
1
+
a
2
+
⋯
+
a
n
0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n
0
,
a
1
,
a
1
+
a
2
,
a
1
+
a
2
+
a
3
,
…
,
a
1
+
a
2
+
⋯
+
a
n
all have di fferent remainders when divided by
n
+
1
n + 1
n
+
1
?
1
1
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Polynomial Equation
Determine all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
(
x
+
1
)
P
(
x
−
1
)
−
(
x
−
1
)
P
(
x
)
(x+1)P(x-1)-(x-1)P(x)
(
x
+
1
)
P
(
x
−
1
)
−
(
x
−
1
)
P
(
x
)
is a constant polynomial.