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National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
1997 Switzerland Team Selection Test
1997 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(4)
4
1
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swiss tst
4. Let
v
v
v
and
w
w
w
be two randomly chosen roots of the equation
z
1997
−
1
=
0
z^{1997} -1 = 0
z
1997
−
1
=
0
(all roots are equiprobable). Find the probability that
2
+
3
≤
∣
u
+
w
∣
\sqrt{2+\sqrt{3}}\le |u+w|
2
+
3
≤
∣
u
+
w
∣
3
1
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swiss tst
3. A 6×6 square has been tiled by 18 dominoes. Show that there exists a line that divides the square into two parts, each of which is also tiled by dominoes
2
1
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SWISS tst
2. Let ABCD be a convex quadrilateral. Find the necessary and sufficient condition for the existence of point P inside the quadrilateral such that the triangles ABP,BCP,CDP,DAP have the same area
1
1
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Swiss tst
1. A finite sequence of integers
a
0
,
a
1
,
.
.
.
,
a
n
a_0,a_1,...,a_n
a
0
,
a
1
,
...
,
a
n
is called quadratic if
∣
a
k
−
a
k
−
1
∣
=
k
2
|a_k -a_{k-1}| = k^2
∣
a
k
−
a
k
−
1
∣
=
k
2
for
n
≥
k
≥
1
n\geq k\geq1
n
≥
k
≥
1
. (a) Prove that for any two integers
b
b
b
and
c
c
c
, there exist a natural number
n
n
n
and a quadratic sequence with
a
0
=
b
a_0 = b
a
0
=
b
and
a
n
=
c
a_n =c
a
n
=
c
. (b) Find the smallest natural number
n
n
n
for which there exists a quadratic sequence with
a
0
=
0
a_0 = 0
a
0
=
0
and
a
n
=
1997
a_n = 1997
a
n
=
1997