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Problems
Contests
National and Regional Contests
Italy Contests
Italy TST
2002 Italy TST
2002 Italy TST
Part of
Italy TST
Subcontests
(3)
1
2
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Find all possible values of AC
Given that in a triangle
A
B
C
ABC
A
BC
,
A
B
=
3
AB=3
A
B
=
3
,
B
C
=
4
BC=4
BC
=
4
and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of
A
C
AC
A
C
.
The circumcentre of triangle AED
A scalene triangle
A
B
C
ABC
A
BC
is inscribed in a circle
Γ
\Gamma
Γ
. The bisector of angle
A
A
A
meets
B
C
BC
BC
at
E
E
E
. Let
M
M
M
be the midpoint of the arc
B
A
C
BAC
B
A
C
. The line
M
E
ME
ME
intersects
Γ
\Gamma
Γ
again at
D
D
D
. Show that the circumcentre of triangle
A
E
D
AED
A
E
D
coincides with the intersection point of the tangent to
Γ
\Gamma
Γ
at
D
D
D
and the line
B
C
BC
BC
.
2
2
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Divisiblity with a binomial
Prove that for each prime number
p
p
p
and positive integer
n
n
n
,
p
n
p^n
p
n
divides
(
p
n
p
)
−
p
n
−
1
.
\binom{p^n}{p}-p^{n-1}.
(
p
p
n
)
−
p
n
−
1
.
soccer tournament
On a soccer tournament with
n
≥
3
n\ge 3
n
≥
3
teams taking part, several matches are played in such a way that among any three teams, some two play a match.
(
a
)
(a)
(
a
)
If
n
=
7
n=7
n
=
7
, find the smallest number of matches that must be played.
(
b
)
(b)
(
b
)
Find the smallest number of matches in terms of
n
n
n
.
3
2
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Finitely many x such that f(x)=1 functional equation
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^+\rightarrow\mathbb{R}^+
f
:
R
+
→
R
+
which satisfy the following conditions:
(
i
)
(\text{i})
(
i
)
f
(
x
+
f
(
y
)
)
=
f
(
x
)
f
(
y
)
f(x+f(y))=f(x)f(y)
f
(
x
+
f
(
y
))
=
f
(
x
)
f
(
y
)
for all
x
,
y
>
0
;
x,y>0;
x
,
y
>
0
;
(
ii
)
(\text{ii})
(
ii
)
there are at most finitely many
x
x
x
with
f
(
x
)
=
1
f(x)=1
f
(
x
)
=
1
.
x divides y^2+m
Prove that for any positive integer
m
m
m
there exist an infinite number of pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
(
i
)
(\text{i})
(
i
)
x
x
x
and
y
y
y
are relatively prime;
(
ii
)
(\text{ii})
(
ii
)
x
x
x
divides
y
2
+
m
;
y^2+m;
y
2
+
m
;
(
iii
)
(\text{iii})
(
iii
)
y
y
y
divides
x
2
+
m
.
x^2+m.
x
2
+
m
.