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National and Regional Contests
Italy Contests
Italy TST
1997 Italy TST
1997 Italy TST
Part of
Italy TST
Subcontests
(4)
4
1
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TST italy
There are
n
n
n
pawns on
n
n
n
distinct squares of a
19
×
19
19\times 19
19
×
19
chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?
3
1
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Italy TST
Determine all triples
(
x
,
y
,
p
)
(x,y, p)
(
x
,
y
,
p
)
with
x
x
x
,
y
y
y
positive integers and
p
p
p
a prime number verifying the equation
p
x
−
y
p
=
1
p^x -y^p = 1
p
x
−
y
p
=
1
.
2
1
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italian team selection test
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Suppose that the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
meets the side
A
C
AC
A
C
at point
D
D
D
such that
B
C
=
B
D
+
A
D
BC = BD+AD
BC
=
B
D
+
A
D
. Find the measure of
∠
B
A
C
\angle BAC
∠
B
A
C
.
1
1
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very old problem
Let
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
be real numbers with
x
,
y
,
z
x,y,z
x
,
y
,
z
not all equal such that
x
+
1
y
=
y
+
1
z
=
z
+
1
x
=
t
.
x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.
x
+
y
1
=
y
+
z
1
=
z
+
x
1
=
t
.
Find all possible values of
t
t
t
such that
x
y
z
+
t
=
0
xyz+t=0
x
yz
+
t
=
0
.