MathDB
Problems
Contests
National and Regional Contests
Italy Contests
Italy TST
2006 Italy TST
2006 Italy TST
Part of
Italy TST
Subcontests
(3)
2
2
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INequality involving midpoints and orthocentre
Let
A
B
C
ABC
A
BC
be a triangle, let
H
H
H
be the orthocentre and
L
,
M
,
N
L,M,N
L
,
M
,
N
the midpoints of the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
respectively. Prove that
H
L
2
+
H
M
2
+
H
N
2
<
A
L
2
+
B
M
2
+
C
N
2
HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}
H
L
2
+
H
M
2
+
H
N
2
<
A
L
2
+
B
M
2
+
C
N
2
if and only if
A
B
C
ABC
A
BC
is acute-angled.
Number of a with n|a^n+1
Let
n
n
n
be a positive integer, and let
A
n
A_{n}
A
n
be the the set of all positive integers
a
≤
n
a\le n
a
≤
n
such that
n
∣
a
n
+
1
n|a^{n}+1
n
∣
a
n
+
1
. a) Find all
n
n
n
such that
A
n
≠
∅
A_{n}\neq \emptyset
A
n
=
∅
b) Find all
n
n
n
such that
∣
A
n
∣
|{A_{n}}|
∣
A
n
∣
is even and non-zero. c) Is there
n
n
n
such that
∣
A
n
∣
=
130
|{A_{n}}| = 130
∣
A
n
∣
=
130
?
3
2
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Functional equation from Z to Z
Find all functions
f
:
Z
→
Z
f : \mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
such that for all integers
m
,
n
m,n
m
,
n
,
f
(
m
−
n
+
f
(
n
)
)
=
f
(
m
)
+
f
(
n
)
.
f(m - n + f(n)) = f(m) + f(n).
f
(
m
−
n
+
f
(
n
))
=
f
(
m
)
+
f
(
n
)
.
P(x) has a multiple with real positive coefficients
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with complex coefficients such that
P
(
0
)
≠
0
P(0)\neq 0
P
(
0
)
=
0
. Prove that there exists a multiple of
P
(
x
)
P(x)
P
(
x
)
with real positive coefficients if and only if
P
(
x
)
P(x)
P
(
x
)
has no real positive root.
1
2
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Good strings
Let
S
S
S
be a string of
99
99
99
characters,
66
66
66
of which are
A
A
A
and
33
33
33
are
B
B
B
. We call
S
S
S
good if, for each
n
n
n
such that
1
≤
n
≤
99
1\le n \le 99
1
≤
n
≤
99
, the sub-string made from the first
n
n
n
characters of
S
S
S
has an odd number of distinct permutations. How many good strings are there? Which strings are good?
Parallel tangets
The circles
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
intersect at the points
Q
Q
Q
and
R
R
R
and internally touch a circle
γ
\gamma
γ
at
A
1
A_1
A
1
and
A
2
A_2
A
2
respectively. Let
P
P
P
be an arbitrary point on
γ
\gamma
γ
. Segments
P
A
1
PA_1
P
A
1
and
P
A
2
PA_2
P
A
2
meet
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
again at
B
1
B_1
B
1
and
B
2
B_2
B
2
respectively. a) Prove that the tangent to
γ
1
\gamma_{1}
γ
1
at
B
1
B_{1}
B
1
and the tangent to
γ
2
\gamma_{2}
γ
2
at
B
2
B_{2}
B
2
are parallel. b) Prove that
B
1
B
2
B_{1}B_{2}
B
1
B
2
is the common tangent to
γ
1
\gamma_{1}
γ
1
and
γ
2
\gamma_{2}
γ
2
iff
P
P
P
lies on
Q
R
QR
QR
.