MathDB
Problems
Contests
International Contests
Pan African
2006 Pan African
2006 Pan African
Part of
Pan African
Subcontests
(6)
6
1
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Two incentres collinear with two points of intersection
Let
A
B
C
ABC
A
BC
be a right angled triangle at
A
A
A
. Denote
D
D
D
the foot of the altitude through
A
A
A
and
O
1
,
O
2
O_1, O_2
O
1
,
O
2
the incentres of triangles
A
D
B
ADB
A
D
B
and
A
D
C
ADC
A
D
C
. The circle with centre
A
A
A
and radius
A
D
AD
A
D
cuts
A
B
AB
A
B
in
K
K
K
and
A
C
AC
A
C
in
L
L
L
. Show that
O
1
,
O
2
,
K
O_1, O_2, K
O
1
,
O
2
,
K
and
L
L
L
are on a line.
5
1
Hide problems
Integers divded into 3 sets, with no consecutive integers
In how many ways can the integers from
1
1
1
to
2006
2006
2006
be divided into three non-empty disjoint sets so that none of these sets contains a pair of consecutive integers?
4
1
Hide problems
Determine v_2(1)+v_2(2)+...+v_2(n)
For every positive integer
k
k
k
let
a
(
k
)
a(k)
a
(
k
)
be the largest integer such that
2
a
(
k
)
2^{a(k)}
2
a
(
k
)
divides
k
k
k
. For every positive integer
n
n
n
determine
a
(
1
)
+
a
(
2
)
+
⋯
+
a
(
2
n
)
a(1)+a(2)+\cdots+a(2^n)
a
(
1
)
+
a
(
2
)
+
⋯
+
a
(
2
n
)
.
3
1
Hide problems
Epression is an integer if and only if {a}+{b}+{c} is
For a real number
x
x
x
let
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
be the greatest integer less than or equal to
x
x
x
and let
{
x
}
=
x
−
⌊
x
⌋
\{x\} = x - \lfloor x\rfloor
{
x
}
=
x
−
⌊
x
⌋
. If
a
,
b
,
c
a, b, c
a
,
b
,
c
are distinct real numbers, prove that
a
3
(
a
−
b
)
(
a
−
c
)
+
b
3
(
b
−
a
)
(
b
−
c
)
+
c
3
(
c
−
a
)
(
c
−
b
)
\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}
(
a
−
b
)
(
a
−
c
)
a
3
+
(
b
−
a
)
(
b
−
c
)
b
3
+
(
c
−
a
)
(
c
−
b
)
c
3
is an integer if and only if
{
a
}
+
{
b
}
+
{
c
}
\{a\} + \{b\} + \{c\}
{
a
}
+
{
b
}
+
{
c
}
is an integer.
2
1
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Which integers are (a/b+b/c+c/a) and (b/a+a/c+c/b)?
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be three non-zero integers. It is known that the sums
a
b
+
b
c
+
c
a
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}
b
a
+
c
b
+
a
c
and
b
a
+
c
b
+
a
c
\frac{b}{a}+\frac{c}{b}+\frac{a}{c}
a
b
+
b
c
+
c
a
are integers. Find these sums.
1
1
Hide problems
Two equal segments defined by perpendicular diameters
Let
A
B
AB
A
B
and
C
D
CD
C
D
be two perpendicular diameters of a circle with centre
O
O
O
. Consider a point
M
M
M
on the diameter
A
B
AB
A
B
, different from
A
A
A
and
B
B
B
. The line
C
M
CM
CM
cuts the circle again at
N
N
N
. The tangent at
N
N
N
to the circle and the perpendicular at
M
M
M
to
A
M
AM
A
M
intersect at
P
P
P
. Show that
O
P
=
C
M
OP = CM
OP
=
CM
.