MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1990 Flanders Math Olympiad
1990 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
1
1
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bourgondian cross (I think)
On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=277
3
1
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easy probability
We form a decimal code of
21
21
21
digits. the code may start with
0
0
0
. Determine the probability that the fragment
0123456789
0123456789
0123456789
appears in the code.
4
1
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decreasing function
Let
f
:
R
0
+
→
R
0
+
f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0
f
:
R
0
+
→
R
0
+
be a strictly decreasing function. (a) Be
a
n
a_n
a
n
a sequence of strictly positive reals so that
∀
k
∈
N
0
:
k
⋅
f
(
a
k
)
≥
(
k
+
1
)
⋅
f
(
a
k
+
1
)
\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})
∀
k
∈
N
0
:
k
⋅
f
(
a
k
)
≥
(
k
+
1
)
⋅
f
(
a
k
+
1
)
Prove that
a
n
a_n
a
n
is ascending, that
lim
k
→
+
∞
f
(
a
k
)
\displaystyle\lim_{k\rightarrow +\infty} f(a_k)
k
→
+
∞
lim
f
(
a
k
)
= 0and that
lim
k
→
+
∞
a
k
=
+
∞
\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty
k
→
+
∞
lim
a
k
=
+
∞
(b) Prove that there exist such a sequence (
a
n
a_n
a
n
) in
R
0
+
\mathbb{R}^+_0
R
0
+
if you know
lim
x
→
+
∞
f
(
x
)
=
0
\displaystyle\lim_{x\rightarrow +\infty} f(x)=0
x
→
+
∞
lim
f
(
x
)
=
0
.
2
1
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Too easy for advanced
Let
a
a
a
and
b
b
b
be two primes having at least two digits, such that
a
>
b
a > b
a
>
b
. Show that
240
∣
(
a
4
−
b
4
)
240|\left(a^4-b^4\right)
240∣
(
a
4
−
b
4
)
and show that 240 is the greatest positive integer having this property.