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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1997 Greece National Olympiad
1997 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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Inequality for a polynomial on Z[x]
A polynomial
P
P
P
with integer coefficients has at least
13
13
13
distinct integer roots. Prove that if an integer
n
n
n
is not a root of
P
P
P
, then
∣
P
(
n
)
∣
≥
7
⋅
6
!
2
|P(n)| \geq 7 \cdot 6!^2
∣
P
(
n
)
∣
≥
7
⋅
6
!
2
, and give an example for equality.
1
1
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Extrema for a function defined on a square
Let
P
P
P
be a point inside or on the boundary of a square
A
B
C
D
ABCD
A
BC
D
. Find the minimum and maximum values of
f
(
P
)
=
∠
A
B
P
+
∠
B
C
P
+
∠
C
D
P
+
∠
D
A
P
f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP
f
(
P
)
=
∠
A
BP
+
∠
BCP
+
∠
C
D
P
+
∠
D
A
P
.
2
1
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Find f(1)
Let a function
f
:
R
+
→
R
f : \Bbb{R}^+ \to \Bbb{R}
f
:
R
+
→
R
satisfy: (i)
f
f
f
is strictly increasing, (ii)
f
(
x
)
>
−
1
/
x
f(x) > -1/x
f
(
x
)
>
−
1/
x
for all
x
>
0
x > 0
x
>
0
, (iii)
f
(
x
)
f
(
f
(
x
)
+
1
/
x
)
=
1
f(x)f (f(x) + 1/x) = 1
f
(
x
)
f
(
f
(
x
)
+
1/
x
)
=
1
for all
x
>
0
x > 0
x
>
0
. Determine
f
(
1
)
f(1)
f
(
1
)
.
3
1
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A diophantine equation
Find all integer solutions to
13
x
2
+
1996
y
2
=
z
1997
.
\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.
x
2
13
+
y
2
1996
=
1997
z
.