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Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
1998 Greece JBMO TST
1998 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(5)
5
1
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Greece TST 1998 Q5
Let
I
I
I
be an open interval of length
1
n
\frac{1}{n}
n
1
, where
n
n
n
is a positive integer. Find the maximum possible number of rational numbers of the form
a
b
\frac{a}{b}
b
a
where
1
≤
b
≤
n
1 \le b \le n
1
≤
b
≤
n
that lie in
I
I
I
.
4
1
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Greece TST 1998 Q4
(a) A polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients takes the value
−
2
-2
−
2
for at least seven distinct integers
x
x
x
. Prove that it cannot take the value
1996
1996
1996
. (b) Prove that there are irrational numbers
x
,
y
x,y
x
,
y
such that
x
y
x^y
x
y
is rational.
3
1
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Greece TST 1998 Q3
Prove that if the number
A
=
111
⋯
1
A = 111 \cdots 1
A
=
111
⋯
1
(
n
n
n
digits) is prime, then
n
n
n
is prime. Is the converse true?
2
1
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Greece TST 1998 Q2
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with parallel sides
A
B
,
C
D
AB, CD
A
B
,
C
D
.
M
,
N
M,N
M
,
N
lie on lines
A
D
,
B
C
AD, BC
A
D
,
BC
respectively such that
M
N
∣
∣
A
B
MN || AB
MN
∣∣
A
B
. Prove that
D
C
⋅
M
A
+
A
B
⋅
M
D
=
M
N
⋅
A
D
DC \cdot MA + AB \cdot MD = MN \cdot AD
D
C
⋅
M
A
+
A
B
⋅
M
D
=
MN
⋅
A
D
.
1
1
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Greece TST 1998 Q1
If
x
,
y
,
z
>
0
,
k
>
2
x,y,z > 0, k>2
x
,
y
,
z
>
0
,
k
>
2
and
a
=
x
+
k
y
+
k
z
,
b
=
k
x
+
y
+
k
z
,
c
=
k
x
+
k
y
+
z
a=x+ky+kz, b=kx+y+kz, c=kx+ky+z
a
=
x
+
k
y
+
k
z
,
b
=
k
x
+
y
+
k
z
,
c
=
k
x
+
k
y
+
z
, show that
x
a
+
y
b
+
z
c
≥
3
2
k
+
1
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} \ge \frac{3}{2k+1}
a
x
+
b
y
+
c
z
≥
2
k
+
1
3
.