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Contests
National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1995 Abels Math Contest (Norwegian MO)
1995 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(5)
4
1
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\sum (x_i +y_i)^2 \sum 1/x_iy_i >= 4n^2
Let
x
i
,
y
i
x_i,y_i
x
i
,
y
i
be positive real numbers,
i
=
1
,
2
,
.
.
.
,
n
i = 1,2,...,n
i
=
1
,
2
,
...
,
n
. Prove that
(
∑
i
=
1
n
(
x
i
+
y
i
)
2
)
(
∑
i
=
1
n
1
x
i
y
i
)
≥
4
n
2
\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2
(
i
=
1
∑
n
(
x
i
+
y
i
)
2
)
(
i
=
1
∑
n
x
i
y
i
1
)
≥
4
n
2
3
1
Hide problems
\sum_{i=1}^n \frac{1}{x_i} includes all natural numbers
Show that there exists a sequence
x
1
,
x
2
,
.
.
.
x_1,x_2,...
x
1
,
x
2
,
...
of natural numbers in which every natural number occurs exactly once, such that the sums
∑
i
=
1
n
1
x
i
\sum_{i=1}^n \frac{1}{x_i}
∑
i
=
1
n
x
i
1
,
n
=
1
,
2
,
3
,
.
.
.
n = 1,2,3,...
n
=
1
,
2
,
3
,
...
, include all natural numbers.
2b
1
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point lies on perp. bisector wanted, intersecting circles given
Two circles of the same radii intersect in two distinct points
P
P
P
and
Q
Q
Q
. A line passing through
P
P
P
, not touching any of the circles, intersects the circles again at
A
A
A
and
B
B
B
. Prove that
Q
Q
Q
lies on the perpendicular bisector of
A
B
AB
A
B
.
2a
1
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perpendicularity wanted, starting with tangent circles
Two circles
k
1
,
k
2
k_1,k_2
k
1
,
k
2
touch each other at
P
P
P
, and touch a line
ℓ
\ell
ℓ
at
A
A
A
and
B
B
B
respectively. Line
A
P
AP
A
P
meets
k
2
k_2
k
2
at
C
C
C
. Prove that
B
C
BC
BC
is perpendicular to
ℓ
\ell
ℓ
.
1b
1
Hide problems
(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1 => x+y=0
Prove that if
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
for real numbers
x
,
y
x,y
x
,
y
, then
x
+
y
=
0
x+y = 0
x
+
y
=
0
.