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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2004 Swedish Mathematical Competition
2004 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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every convex n-gon of area 1 contains a quadrilateral of area at least 1/2
Prove that every convex
n
n
n
-gon of area
1
1
1
contains a quadrilateral of area at least
1
2
\frac12
2
1
. .
5
1
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n-1 lines in nxn board
A square of side
n
≥
2
n \ge 2
n
≥
2
is divided into
n
2
n^2
n
2
unit squares (
n
∈
N
n \in N
n
∈
N
). One draws
n
−
1
n-1
n
−
1
lines so that the interior of each of the unit squares is cut by at least one of these lines. (a) Give an example of such a configuration for some
n
n
n
. (b) Show that some two of the lines must meet inside the square.
4
1
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sinv < 20/21 ? if tan v = 2v and 0 < v < \pi / 2
If
0
<
v
<
π
2
0 < v <\frac{\pi}{2}
0
<
v
<
2
π
and
tan
v
=
2
v
\tan v = 2v
tan
v
=
2
v
, decide whether
s
i
n
v
<
20
21
sinv < \frac{20}{21}
s
in
v
<
21
20
.
3
1
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f(x)+x f(1-x) = x^2
A function
f
f
f
satisfies
f
(
x
)
+
x
f
(
1
−
x
)
=
x
2
f(x)+x f(1-x) = x^2
f
(
x
)
+
x
f
(
1
−
x
)
=
x
2
for all real
x
x
x
. Determine
f
f
f
.
2
1
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100 coins in a bag in acountry where coins exist of 1,2,3,4 ,5.
In one country there are coins of value
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
or
5
5
5
. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has
100
100
100
coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?
1
1
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common area of 2 equal circles intersecting at a right angle
Two circles in the plane, both of radius
R
R
R
, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.