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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2021 Flanders Math Olympiad
2021 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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f(x/(x^2 + x + 1))=x^2/(x^4 + x^2 + 1)
(a) Prove that for every
x
∈
R
x \in R
x
∈
R
holds that
−
1
≤
x
x
2
+
x
+
1
≤
1
3
-1 \le \frac{x}{x^2 + x + 1} \le \frac 13
−
1
≤
x
2
+
x
+
1
x
≤
3
1
(b) Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
for which for every
x
∈
R
x \in R
x
∈
R
holds that
f
(
x
x
2
+
x
+
1
)
=
x
2
x
4
+
x
2
+
1
f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}
f
(
x
2
+
x
+
1
x
)
=
x
4
+
x
2
+
1
x
2
1
1
Hide problems
is 100 000 a plum number
Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number
2
2
2
. After that, Jantje picks the plum each time with the smallest number
n
n
n
that satisfies the following two conditions:
∙
\bullet
∙
n
n
n
is greater than all numbers on the already picked plums,
∙
\bullet
∙
n
n
n
is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is
100000
100 000
100000
a plum number? Justify your answer.
3
1
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19 numbered balls in a box, we pick 5 different balls
There are
19
19
19
balls in a box, numbered
1
1
1
through
19
19
19
. When we go out get that box without looking five different balls, which number has the largest probability of being the difference between the highest and lowest number drawn? Justify you reply .
2
1
Hide problems
5 congruent circles in a regular pentagon
Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of
56
56
56
centimeters and the second one of
50
50
50
centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area?https://cdn.artofproblemsolving.com/attachments/1/0/68e80530742f1f0775aff5a265e0c9928fa66c.png