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International Contests
Nordic
2011 Nordic
2011 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Harmonic sums of relatively prime integers
Show that for any integer
n
≥
2
n \ge 2
n
≥
2
the sum of the fractions
1
a
b
\frac{1}{ab}
ab
1
, where
a
a
a
and
b
b
b
are relatively prime positive integers such that
a
<
b
≤
n
a < b \le n
a
<
b
≤
n
and
a
+
b
>
n
a+b > n
a
+
b
>
n
, equals
1
2
\frac{1}{2}
2
1
. (Integers
a
a
a
and
b
b
b
are called relatively prime if the greatest common divisor of
a
a
a
and
b
b
b
is
1
1
1
.)
3
1
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Functional: f(f(x)+y)=f(x^2-y)+4yf(x)
Find all functions
f
f
f
such that
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
y
f
(
x
)
f(f(x) + y) = f(x^2-y) + 4yf(x)
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
y
f
(
x
)
for all real numbers
x
x
x
and
y
y
y
.
2
1
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Ratio between parallel segments in an isosceles triangle
In a triangle
A
B
C
ABC
A
BC
assume
A
B
=
A
C
AB = AC
A
B
=
A
C
, and let
D
D
D
and
E
E
E
be points on the extension of segment
B
A
BA
B
A
beyond
A
A
A
and on the segment
B
C
BC
BC
, respectively, such that the lines
C
D
CD
C
D
and
A
E
AE
A
E
are parallel. Prove CD \ge \frac{4h}{BC}CE, where
h
h
h
is the height from
A
A
A
in triangle
A
B
C
ABC
A
BC
. When does equality hold?
1
1
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Can a number plus its reverse have only odd digits
When
a
0
,
a
1
,
…
,
a
1000
a_0, a_1, \dots , a_{1000}
a
0
,
a
1
,
…
,
a
1000
denote digits, can the sum of the
1001
1001
1001
-digit numbers
a
0
a
1
⋯
a
1000
a_0a_1\cdots a_{1000}
a
0
a
1
⋯
a
1000
and
a
1000
a
999
⋯
a
0
a_{1000}a_{999}\cdots a_0
a
1000
a
999
⋯
a
0
have odd digits only?