MathDB
Problems
Contests
International Contests
Pan African
2000 Pan African
2000 Pan African
Part of
Pan African
Subcontests
(3)
3
2
Hide problems
p is divisible by 2003
Let
p
p
p
and
q
q
q
be coprime positive integers such that:
p
q
=
1
−
1
2
+
1
3
−
1
4
⋯
−
1
1334
+
1
1335
\dfrac{p}{q}=1-\frac12+\frac13-\frac14 \cdots -\dfrac{1}{1334}+\dfrac{1}{1335}
q
p
=
1
−
2
1
+
3
1
−
4
1
⋯
−
1334
1
+
1335
1
Prove
p
p
p
is divisible by 2003.
Keys and locks
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers
n
n
n
such that it is possible to give each of the directors a set of keys to
n
n
n
different locks, according to the requirements and regulations of the company.
2
2
Hide problems
Polynomial
Define the polynomials
P
0
,
P
1
,
P
2
⋯
P_0, P_1, P_2 \cdots
P
0
,
P
1
,
P
2
⋯
by:
P
0
(
x
)
=
x
3
+
213
x
2
−
67
x
−
2000
P_0(x)=x^3+213x^2-67x-2000
P
0
(
x
)
=
x
3
+
213
x
2
−
67
x
−
2000
P
n
(
x
)
=
P
n
−
1
(
x
−
n
)
,
n
∈
N
P_n(x)=P_{n-1}(x-n), n \in N
P
n
(
x
)
=
P
n
−
1
(
x
−
n
)
,
n
∈
N
Find the coefficient of
x
x
x
in
P
21
(
x
)
P_{21}(x)
P
21
(
x
)
.
Circle and Tangent lines
Let
γ
\gamma
γ
be circle and let
P
P
P
be a point outside
γ
\gamma
γ
. Let
P
A
PA
P
A
and
P
B
PB
PB
be the tangents from
P
P
P
to
γ
\gamma
γ
(where
A
,
B
∈
γ
A, B \in \gamma
A
,
B
∈
γ
). A line passing through
P
P
P
intersects
γ
\gamma
γ
at points
Q
Q
Q
and
R
R
R
. Let
S
S
S
be a point on
γ
\gamma
γ
such that
B
S
∥
Q
R
BS \parallel QR
BS
∥
QR
. Prove that
S
A
SA
S
A
bisects
Q
R
QR
QR
.
1
2
Hide problems
Trig Equation
Solve for
x
∈
R
x \in R
x
∈
R
:
sin
3
x
(
1
+
cot
x
)
+
cos
3
x
(
1
+
tan
x
)
=
cos
2
x
\sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x}
sin
3
x
(
1
+
cot
x
)
+
cos
3
x
(
1
+
tan
x
)
=
cos
2
x
System of equations
Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers such that
a
2
+
b
2
=
c
2
a^2+b^2=c^2
a
2
+
b
2
=
c
2
, solve the system:
z
2
=
x
2
+
y
2
z^2=x^2+y^2
z
2
=
x
2
+
y
2
(
z
+
c
)
2
=
(
x
+
a
)
2
+
(
y
+
b
)
2
(z+c)^2=(x+a)^2+(y+b)^2
(
z
+
c
)
2
=
(
x
+
a
)
2
+
(
y
+
b
)
2
in real numbers
x
,
y
x, y
x
,
y
and
z
z
z
.