MathDB
Problems
Contests
International Contests
Pan African
2013 Pan African
2013 Pan African
Part of
Pan African
Subcontests
(3)
3
2
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Hexagon with an equal area distribution
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon with
∠
A
=
∠
D
\angle A= \angle D
∠
A
=
∠
D
and
∠
B
=
∠
E
\angle B=\angle E
∠
B
=
∠
E
. Let
K
K
K
and
L
L
L
be the midpoints of the sides
A
B
AB
A
B
and
D
E
DE
D
E
respectively. Prove that the sum of the areas of triangles
F
A
K
FAK
F
A
K
,
K
C
B
KCB
K
CB
and
C
F
L
CFL
CF
L
is equal to half of the area of the hexagon if and only if
B
C
C
D
=
E
F
F
A
.
\frac{BC}{CD}=\frac{EF}{FA}.
C
D
BC
=
F
A
EF
.
Solve the system of inequalities
Let
x
x
x
,
y
y
y
, and
z
z
z
be real numbers such that
x
<
y
<
z
<
6
x<y<z<6
x
<
y
<
z
<
6
. Solve the system of inequalities:
{
1
y
−
x
+
1
z
−
y
≤
2
1
6
−
z
+
2
≤
x
\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\ \dfrac{1}{6-z}+2\le x \\ \end{array}\right.
⎩
⎨
⎧
y
−
x
1
+
z
−
y
1
≤
2
6
−
z
1
+
2
≤
x
2
2
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Functional: f(x)f(y)+f(x+y)=xy
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
+
f
(
x
+
y
)
=
x
y
f(x)f(y)+f(x+y)=xy
f
(
x
)
f
(
y
)
+
f
(
x
+
y
)
=
x
y
for all real numbers
x
x
x
and
y
y
y
.
Every 3x3 square of an nxn board has such a colouring
The cells of an
n
×
n
n\times n
n
×
n
board with
n
≥
5
n\ge 5
n
≥
5
are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any
3
×
3
3\times 3
3
×
3
square within the board, two of its corner squares are coloured black and two are coloured white.
1
2
Hide problems
An integer for which n(n+2013) is a square
A positive integer
n
n
n
is such that
n
(
n
+
2013
)
n(n+2013)
n
(
n
+
2013
)
is a perfect square. a) Show that
n
n
n
cannot be prime. b) Find a value of
n
n
n
such that
n
(
n
+
2013
)
n(n+2013)
n
(
n
+
2013
)
is a perfect square.
Equal sets of angles in a trapezium
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
B
AB
A
B
parallel to
C
D
CD
C
D
. Let
P
P
P
and
Q
Q
Q
be the midpoints of
A
C
AC
A
C
and
B
D
BD
B
D
, respectively. Prove that if
∠
A
B
P
=
∠
C
B
D
\angle ABP=\angle CBD
∠
A
BP
=
∠
CB
D
, then
∠
B
C
Q
=
∠
A
C
D
\angle BCQ=\angle ACD
∠
BCQ
=
∠
A
C
D
.