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National and Regional Contests
Slovenia Contests
Slovenia National Olympiad
1997 Slovenia National Olympiad
1997 Slovenia National Olympiad
Part of
Slovenia National Olympiad
Subcontests
(4)
Problem 4
4
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Problem 3
4
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Problem 2
4
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Problem 1
3
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k=m+2mn+n iff 2k+1 composite (Slovenia 1997 1st Grade P1)
Let
k
k
k
be a positive integer. Prove that: (a) If
k
=
m
+
2
m
n
+
n
k=m+2mn+n
k
=
m
+
2
mn
+
n
for some positive integers
m
,
n
m,n
m
,
n
, then
2
k
+
1
2k+1
2
k
+
1
is composite. (b) If
2
k
+
1
2k+1
2
k
+
1
is composite, then there exist positive integers
m
,
n
m,n
m
,
n
such that
k
=
m
+
2
m
n
+
n
k=m+2mn+n
k
=
m
+
2
mn
+
n
.
m+n-1|m^2+n^2-1 implies m+n-1 composite (Slovenia 1997 3rd Grade P1)
Suppose that
m
,
n
m,n
m
,
n
are integers greater than
1
1
1
such that
m
+
n
−
1
m+n-1
m
+
n
−
1
divides
m
2
+
n
2
−
1
m^2+n^2-1
m
2
+
n
2
−
1
. Prove that
m
+
n
−
1
m+n-1
m
+
n
−
1
cannot be a prime number.
p+q-pq=154, p,q prime (Slovenia 1997 4th Grade P1)
Marko chose two prime numbers
a
a
a
and
b
b
b
with the same number of digits and wrote them down one after another, thus obtaining a number
c
c
c
. When he decreased
c
c
c
by the product of
a
a
a
and
b
b
b
, he got the result
154
154
154
. Determine the number
c
c
c
.