MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2015 Flanders Math Olympiad
2015 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
Hide problems
sum is product
Show that for
n
≥
5
n \geq 5
n
≥
5
, the integers
1
,
2
,
…
n
1, 2, \ldots n
1
,
2
,
…
n
can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.
3
1
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squares and busses
A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is
1
2
\frac{1}{2}
2
1
. a) Show that the number of people in the group is a square. b) Show that the number of seats on each bus is a triangular number.
2
1
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Simple geometry
Consider two points
Y
Y
Y
and
X
X
X
in a plane and a variable point
P
P
P
which is not on
X
Y
XY
X
Y
. Let the parallel line to
Y
P
YP
Y
P
through
X
X
X
intersect the internal angle bisector of
∠
X
Y
P
\angle XYP
∠
X
Y
P
in
A
A
A
, and let the parallel line to
X
P
XP
XP
through
Y
Y
Y
intersect the internal angle bisector of
∠
Y
X
P
\angle YXP
∠
Y
XP
in
B
B
B
. Let
A
B
AB
A
B
intersect
X
P
XP
XP
and
Y
P
YP
Y
P
in
S
S
S
and
T
T
T
respectively. Show that the product
∣
X
S
∣
∗
∣
Y
T
∣
|XS|*|YT|
∣
XS
∣
∗
∣
Y
T
∣
does not depend on the position of
P
P
P
.
1
1
Hide problems
labelling the Pentagon
The sides and vertices of a pentagon are labelled with the numbers
1
1
1
through
10
10
10
so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?