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Contests
National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2013 Canadian Mathematical Olympiad Qualification Repechage
2013 Canadian Mathematical Olympiad Qualification Repechage
Part of
Canadian Mathematical Olympiad Qualification Repechage
Subcontests
(8)
8
1
Hide problems
Prove that HA'+HB'+HC'<2R
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute-angled triangle with orthocentre
H
H
H
and circumcentre
O
O
O
. Let
R
R
R
be the radius of the circumcircle. \begin{align*} \text{Let }\mathit{A'}\text{ be the point on }\mathit{AO}\text{ (extended if necessary) for which }\mathit{HA'}\perp\mathit{AO}. \\ \text{Let }\mathit{B'}\text{ be the point on }\mathit{BO}\text{ (extended if necessary) for which }\mathit{HB'}\perp\mathit{BO}. \\ \text{Let }\mathit{C'}\text{ be the point on }\mathit{CO}\text{ (extended if necessary) for which }\mathit{HC'}\perp\mathit{CO}.\end{align*} Prove that
H
A
′
+
H
B
′
+
H
C
′
<
2
R
HA'+HB'+HC'<2R
H
A
′
+
H
B
′
+
H
C
′
<
2
R
(Note: The orthocentre of a triangle is the intersection of the three altitudes of the triangle. The circumcircle of a triangle is the circle passing through the triangle’s three vertices. The circummcentre is the centre of the circumcircle.)
7
1
Hide problems
Triangle Friendly Sequences
Consider the following layouts of nine triangles with the letters
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
A, B, C, D, E, F, G, H, I
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
in its interior.[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(200); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 1.740000000000003, xmax = 8.400000000000013, ymin = 3.500000000000005, ymax = 9.360000000000012; /* image dimensions */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)--cycle); /* draw figures */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)); draw((2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)); draw((7.461947712046029,4.569577506690286)--(5.020000000000005,8.820000000000011)); draw((3.382989341689345,5.990838871467448)--(4.193333333333338,4.580000000000005)); draw((4.202511849578174,7.405966442513598)--(5.828619600041468,4.573707435672692)); draw((5.841878190157451,7.408513542990484)--(4.193333333333338,4.580000000000005)); draw((6.656214943659867,5.990342259816768)--(5.828619600041468,4.573707435672692)); draw((4.202511849578174,7.405966442513598)--(5.841878190157451,7.408513542990484)); draw((3.382989341689345,5.990838871467448)--(6.656214943659867,5.990342259816768)); label("
A
",(4.840000000000007,8.020000000000010),SE*labelscalefactor,fontsize(22)); label("
B
",(3.980000000000006,6.640000000000009),SE*labelscalefactor,fontsize(22)); label("
C
",(4.820000000000007,7.000000000000010),SE*labelscalefactor,fontsize(22)); label("
D
",(5.660000000000008,6.580000000000008),SE*labelscalefactor,fontsize(22)); label("
E
",(3.160000000000005,5.180000000000006),SE*labelscalefactor,fontsize(22)); label("
F
",(4.020000000000006,5.600000000000008),SE*labelscalefactor,fontsize(22)); label("
G
",(4.800000000000007,5.200000000000007),SE*labelscalefactor,fontsize(22)); label("
H
",(5.680000000000009,5.620000000000007),SE*labelscalefactor,fontsize(22)); label("
I
",(6.460000000000010,5.140000000000006),SE*labelscalefactor,fontsize(22)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]A sequence of letters, each letter chosen from
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
A, B, C, D, E, F, G, H, I
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
is said to be triangle-friendly if the first and last letter of the sequence is
C
C
C
, and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after
C
C
C
must be either
A
,
B
A, B
A
,
B
, or
D
D
D
. For example,
C
B
F
B
C
CBF BC
CBFBC
is triangle-friendly, but
C
B
F
G
H
CBF GH
CBFG
H
and
C
B
B
H
C
CBBHC
CBB
H
C
are not.[*] (a) Determine the number of triangle-friendly sequences with
2012
2012
2012
letters.[*] (b) Determine the number of triangle-friendly sequences with exactly
2013
2013
2013
letters.
6
1
Hide problems
Find min/max of x+y+z-xy-yz-zx
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers that are greater than or equal to
0
0
0
and less than or equal to
1
2
\frac{1}{2}
2
1
[*] (a) Determine the minimum possible value of
x
+
y
+
z
−
x
y
−
y
z
−
z
x
x+y+z-xy-yz-zx
x
+
y
+
z
−
x
y
−
yz
−
z
x
and determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
for which this minimum is obtained.[*] (b) Determine the maximum possible value of
x
+
y
+
z
−
x
y
−
y
z
−
z
x
x+y+z-xy-yz-zx
x
+
y
+
z
−
x
y
−
yz
−
z
x
and determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
for which this maximum is obtained.
5
1
Hide problems
S(n)-(5n)=2013
For each positive integer
k
k
k
, let
S
(
k
)
S(k)
S
(
k
)
be the sum of its digits. For example,
S
(
21
)
=
3
S(21) = 3
S
(
21
)
=
3
and
S
(
105
)
=
6
S(105) = 6
S
(
105
)
=
6
. Let
n
n
n
be the smallest integer for which
S
(
n
)
−
S
(
5
n
)
=
2013
S(n) - S(5n) = 2013
S
(
n
)
−
S
(
5
n
)
=
2013
. Determine the number of digits in
n
n
n
.
4
1
Hide problems
Derangements with probability
Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that both of these events occur:[*] (i) Each person receives exactly one gift;[*] (ii) No two people exchanged presents with each other (i.e., if
A
A
A
gave his gift to
B
B
B
, then
B
B
B
did not give her gift to
A
A
A
).
3
1
Hide problems
lcm(y,z)=n
A positive integer
n
n
n
has the property that there are three positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
such that
lcm
(
x
,
y
)
=
180
\text{lcm}(x, y) = 180
lcm
(
x
,
y
)
=
180
,
lcm
(
x
,
z
)
=
900
\text{lcm}(x, z) = 900
lcm
(
x
,
z
)
=
900
, and
lcm
(
y
,
z
)
=
n
\text{lcm}(y, z) = n
lcm
(
y
,
z
)
=
n
, where
lcm
\text{lcm}
lcm
denotes the lowest common multiple. Determine the number of positive integers
n
n
n
with this property.
2
1
Hide problems
Prove that BE bisects <B
In triangle
A
B
C
ABC
A
BC
,
∠
A
=
9
0
∘
\angle A = 90^\circ
∠
A
=
9
0
∘
and
∠
C
=
7
0
∘
\angle C = 70^\circ
∠
C
=
7
0
∘
.
F
F
F
is point on
A
B
AB
A
B
such that
∠
A
C
F
=
3
0
∘
\angle ACF = 30^\circ
∠
A
CF
=
3
0
∘
, and
E
E
E
is a point on
C
A
CA
C
A
such that
∠
C
F
E
=
2
0
∘
\angle CF E = 20^\circ
∠
CFE
=
2
0
∘
. Prove that
B
E
BE
BE
bisects
∠
B
\angle B
∠
B
.
1
1
Hide problems
Find all real solutions
Determine all real solutions to the following equation:
2
(
2
x
)
−
3
⋅
2
(
2
x
−
1
+
1
)
+
8
=
0.
2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.
2
(
2
x
)
−
3
⋅
2
(
2
x
−
1
+
1
)
+
8
=
0.