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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2018 Flanders Math Olympiad
2018 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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3-digit number wanted
Determine all three-digit numbers N such that
N
2
N^2
N
2
has six digits and so that the sum of the number formed by the first three digits of
N
2
N^2
N
2
and the number formed by the latter three digits of
N
2
N^2
N
2
equals
N
N
N
.
3
1
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sums with greatest odd divisor of n
Write down
f
(
n
)
f(n)
f
(
n
)
for the greatest odd divisor of
n
∈
N
0
n \in N_0
n
∈
N
0
. (a) Determine
f
(
n
+
1
)
+
f
(
n
+
2
)
+
.
.
.
+
f
(
2
n
)
f (n + 1) + f (n + 2) + ... + f(2n)
f
(
n
+
1
)
+
f
(
n
+
2
)
+
...
+
f
(
2
n
)
. (b) Determine
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
.
.
.
+
f
(
2
n
)
f(1) + f(2) + f(3) + ... + f(2n)
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
...
+
f
(
2
n
)
.
2
1
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sin ( cos a) < cos( sin a) for acute a
Prove that for every acute angle
α
\alpha
α
,
sin
(
cos
α
)
<
cos
(
sin
α
)
\sin (\cos \alpha) < \cos(\sin \alpha)
sin
(
cos
α
)
<
cos
(
sin
α
)
.
1
1
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<C> 60^o if AB^3 = AC^3 +BC^3 (2018 VWO Flanders MO p1)
In the triangle
△
A
B
C
\vartriangle ABC
△
A
BC
we have
∣
A
B
∣
3
=
∣
A
C
∣
3
+
∣
B
C
∣
3
| AB |^3 = | AC |^3 + | BC |^3
∣
A
B
∣
3
=
∣
A
C
∣
3
+
∣
BC
∣
3
. Prove that
∠
C
>
6
0
o
\angle C> 60^o
∠
C
>
6
0
o
.