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Problems
Contests
International Contests
Nordic
1995 Nordic
1995 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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triangles with sides consecutive integers & area integer
Show that there exist infinitely many mutually non- congruent triangles
T
T
T
, satisfying (i) The side lengths of
T
T
T
are consecutive integers. (ii) The area of
T
T
T
is an integer.
3
1
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Σx_1^2=1, inequality with max{x_i}
Let
n
≥
2
n \ge 2
n
≥
2
and let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
be real numbers satisfying
x
1
+
x
2
+
.
.
.
+
x
n
≥
0
x_1 +x_2 +...+x_n \ge 0
x
1
+
x
2
+
...
+
x
n
≥
0
and
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
1
x_1^2+x_2^2+...+x_n^2=1
x
1
2
+
x
2
2
+
...
+
x
n
2
=
1
. Let
M
=
m
a
x
{
x
1
,
x
2
,
.
.
.
,
x
n
}
M = max \{x_1, x_2,... , x_n\}
M
=
ma
x
{
x
1
,
x
2
,
...
,
x
n
}
. Show that
M
≥
1
n
(
n
−
1
)
M \ge \frac{1}{\sqrt{n(n-1)}}
M
≥
n
(
n
−
1
)
1
(1) .When does equality hold in (1)?
2
1
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number of sequences consisting of 12 numbers with 0s and 1s
Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence
011001
011001
011001
is allowed, but
011101
011101
011101
is not.) Determine the number of sequences consisting of exactly
12
12
12
numbers.
1
1
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equal segments from perpendiculars and a diameter
Let
A
B
AB
A
B
be a diameter of a circle with centre
O
O
O
. We choose a point
C
C
C
on the circumference of the circle such that
O
C
OC
OC
and
A
B
AB
A
B
are perpendicular to each other. Let
P
P
P
be an arbitrary point on the (smaller) arc
B
C
BC
BC
and let the lines
C
P
CP
CP
and
A
B
AB
A
B
meet at
Q
Q
Q
. We choose
R
R
R
on
A
P
AP
A
P
so that
R
Q
RQ
RQ
and
A
B
AB
A
B
are perpendicular to each other. Show that
B
Q
=
Q
R
BQ =QR
BQ
=
QR
.