MathDB
Problems
Contests
International Contests
Pan African
2007 Pan African
2007 Pan African
Part of
Pan African
Subcontests
(3)
3
2
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Make a round-trip of length not divisible by 3
In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by
3
3
3
. (Not all towns need to be visited.)
Greatest area made by perpendiculars through centroid
An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?
2
2
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Locus of points preserving some angles
Let
A
A
A
,
B
B
B
and
C
C
C
be three fixed points, not on the same line. Consider all triangles
A
B
′
C
′
AB'C'
A
B
′
C
′
where
B
′
B'
B
′
moves on a given straight line (not containing
A
A
A
), and
C
′
C'
C
′
is determined such that
∠
B
′
=
∠
B
\angle B'=\angle B
∠
B
′
=
∠
B
and
∠
C
′
=
∠
C
\angle C'=\angle C
∠
C
′
=
∠
C
. Find the locus of
C
′
C'
C
′
.
For which n is this divisible by 2007?
For which positive integers
n
n
n
is
23
1
n
−
22
2
n
−
8
n
−
1
231^n-222^n-8 ^n -1
23
1
n
−
22
2
n
−
8
n
−
1
divisible by
2007
2007
2007
?
1
2
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Sum of digits of N and N+1
Find all natural numbers
N
N
N
consisting of exactly
1112
1112
1112
digits (in decimal notation) such that: (a) The sum of the digits of
N
N
N
is divisible by
2000
2000
2000
; (b) The sum of the digits of
N
+
1
N+1
N
+
1
is divisible by
2000
2000
2000
; (c)
1
1
1
is a digit of
N
N
N
.
System of equations involving square roots
Solve the following system of equations for real
x
,
y
x,y
x
,
y
and
z
z
z
: \begin{eqnarray*} x &=& \sqrt{2y+3}\\ y &=& \sqrt{2z+3}\\ z &=& \sqrt{2x+3}. \end{eqnarray*}