MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1969 Canada National Olympiad
1969 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(10)
10
1
Hide problems
Isosceles right triangle
Let
A
B
C
ABC
A
BC
be the right-angled isosceles triangle whose equal sides have length 1.
P
P
P
is a point on the hypotenuse, and the feet of the perpendiculars from
P
P
P
to the other sides are
Q
Q
Q
and
R
R
R
. Consider the areas of the triangles
A
P
Q
APQ
A
PQ
and
P
B
R
PBR
PBR
, and the area of the rectangle
Q
C
R
P
QCRP
QCRP
. Prove that regardless of how
P
P
P
is chosen, the largest of these three areas is at least
2
/
9
2/9
2/9
.
9
1
Hide problems
Cyclic
Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to
2
\sqrt{2}
2
.
7
1
Hide problems
Easy diophantine
Show that there are no integers
a
,
b
,
c
a,b,c
a
,
b
,
c
for which
a
2
+
b
2
−
8
c
=
6
a^2+b^2-8c=6
a
2
+
b
2
−
8
c
=
6
.
6
1
Hide problems
Classic factorial sum (posted before?)
Find the sum of
1
⋅
1
!
+
2
⋅
2
!
+
3
⋅
3
!
+
⋯
+
(
n
−
1
)
(
n
−
1
)
!
+
n
⋅
n
!
1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!
1
⋅
1
!
+
2
⋅
2
!
+
3
⋅
3
!
+
⋯
+
(
n
−
1
)
(
n
−
1
)!
+
n
⋅
n
!
, where
n
!
=
n
(
n
−
1
)
(
n
−
2
)
⋯
2
⋅
1
n!=n(n-1)(n-2)\cdots2\cdot1
n
!
=
n
(
n
−
1
)
(
n
−
2
)
⋯
2
⋅
1
.
5
1
Hide problems
Angle bisector
Let
A
B
C
ABC
A
BC
be a triangle with sides of length
a
a
a
,
b
b
b
and
c
c
c
. Let the bisector of the angle
C
C
C
cut
A
B
AB
A
B
in
D
D
D
. Prove that the length of
C
D
CD
C
D
is
2
a
b
cos
C
2
a
+
b
.
\frac{2ab\cos \frac{C}{2}}{a+b}.
a
+
b
2
ab
cos
2
C
.
4
1
Hide problems
Perpendiculars
Let
A
B
C
ABC
A
BC
be an equilateral triangle, and
P
P
P
be an arbitrary point within the triangle. Perpendiculars
P
D
,
P
E
,
P
F
PD,PE,PF
P
D
,
PE
,
PF
are drawn to the three sides of the triangle. Show that, no matter where
P
P
P
is chosen,
P
D
+
P
E
+
P
F
A
B
+
B
C
+
C
A
=
1
2
3
.
\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}.
A
B
+
BC
+
C
A
P
D
+
PE
+
PF
=
2
3
1
.
3
1
Hide problems
Hypotenuse inequality
Let
c
c
c
be the length of the hypotenuse of a right angle triangle whose two other sides have lengths
a
a
a
and
b
b
b
. Prove that
a
+
b
≤
c
2
a+b\le c\sqrt{2}
a
+
b
≤
c
2
. When does the equality hold?
2
1
Hide problems
Which is larger?
Determine which of the two numbers
c
+
1
−
c
\sqrt{c+1}-\sqrt{c}
c
+
1
−
c
,
c
−
c
−
1
\sqrt{c}-\sqrt{c-1}
c
−
c
−
1
is greater for any
c
≥
1
c\ge 1
c
≥
1
.
1
1
Hide problems
Odd equation (with equality condition of cauchy)
If
a
1
/
b
1
=
a
2
/
b
2
=
a
3
/
b
3
a_1/b_1=a_2/b_2=a_3/b_3
a
1
/
b
1
=
a
2
/
b
2
=
a
3
/
b
3
and
p
1
,
p
2
,
p
3
p_1,p_2,p_3
p
1
,
p
2
,
p
3
are not all zero, show that for all
n
∈
N
n\in\mathbb{N}
n
∈
N
,
(
a
1
b
1
)
n
=
p
1
a
1
n
+
p
2
a
2
n
+
p
3
a
3
n
p
1
b
1
n
+
p
2
b
2
n
+
p
3
b
3
n
.
\left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}.
(
b
1
a
1
)
n
=
p
1
b
1
n
+
p
2
b
2
n
+
p
3
b
3
n
p
1
a
1
n
+
p
2
a
2
n
+
p
3
a
3
n
.
8
1
Hide problems
Functional equation
Let
f
f
f
be a function with the following properties:1)
f
(
n
)
f(n)
f
(
n
)
is defined for every positive integer
n
n
n
; 2)
f
(
n
)
f(n)
f
(
n
)
is an integer; 3)
f
(
2
)
=
2
f(2)=2
f
(
2
)
=
2
; 4)
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn)=f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
for all
m
m
m
and
n
n
n
; 5)
f
(
m
)
>
f
(
n
)
f(m)>f(n)
f
(
m
)
>
f
(
n
)
whenever
m
>
n
m>n
m
>
n
.Prove that
f
(
n
)
=
n
f(n)=n
f
(
n
)
=
n
.