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Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2016 Flanders Math Olympiad
2016 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
2
1
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min n such that N^n not a divisor of 2016!
Determine the smallest natural number
n
n
n
such that
n
n
n^n
n
n
is not a divisor of the product
1
⋅
2
⋅
3
⋅
.
.
.
⋅
2015
⋅
2016
1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016
1
⋅
2
⋅
3
⋅
...
⋅
2015
⋅
2016
.
4
1
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unique polynomial with pos. integer coeeficients (1) = 6, f(2) = 2016
Prove that there exists a unique polynomial function f with positive integer coefficients such that
f
(
1
)
=
6
f(1) = 6
f
(
1
)
=
6
and
f
(
2
)
=
2016
f(2) = 2016
f
(
2
)
=
2016
.
1
1
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isosceles wanted, AD//BC. 2 circumcircles related
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
is
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
and the angles
∠
A
\angle A
∠
A
and
∠
D
\angle D
∠
D
are acute. The diagonals intersect in
P
P
P
. The circumscribed circles of
△
A
B
P
\vartriangle ABP
△
A
BP
and
△
C
D
P
\vartriangle CDP
△
C
D
P
intersect the line
A
D
AD
A
D
again at
S
S
S
and
T
T
T
respectively. Call
M
M
M
the midpoint of
[
S
T
]
[ST]
[
ST
]
. Prove that
△
B
C
M
\vartriangle BCM
△
BCM
is isosceles. https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png
3
1
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3 line segments divide a triangle into 5 triangles, inequality area problem
Three line segments divide a triangle into five triangles. The area of these triangles is called
u
,
v
,
x
,
u, v, x,
u
,
v
,
x
,
yand
z
z
z
, as in the figure. (a) Prove that
u
v
=
y
z
uv = yz
uv
=
yz
. (b) Prove that the area of the great triangle is at most
x
z
y
\frac{xz}{y}
y
x
z
https://cdn.artofproblemsolving.com/attachments/9/4/2041d62d014cf742876e01dd8c604c4d38a167.png