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Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2011 Grand Duchy of Lithuania
2011 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(5)
2
1
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a_n < 1/(n-1) if (a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}
Let
n
≥
2
n \ge 2
n
≥
2
be a natural number and suppose that positive numbers
a
0
,
a
1
,
.
.
.
,
a
n
a_0,a_1,...,a_n
a
0
,
a
1
,
...
,
a
n
satisfy the equality
(
a
k
−
1
+
a
k
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}
(
a
k
−
1
+
a
k
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
for each
k
=
1
,
2
,
.
.
.
,
n
−
1
k =1,2,...,n -1
k
=
1
,
2
,
...
,
n
−
1
. Prove that
a
n
<
1
n
−
1
a_n< \frac{1}{n-1}
a
n
<
n
−
1
1
5
1
Hide problems
replace 2 numbers by least prime divisor of their sum
Positive integers
1
,
2
,
3
,
.
.
.
,
n
1, 2, 3, ..., n
1
,
2
,
3
,
...
,
n
are written on a blackboard (
n
>
2
n > 2
n
>
2
). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number
97
97
97
remains. Find the least
n
n
n
for which it is possible.
3
1
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p^3 - q^7 = p - q
Find all primes
p
,
q
p,q
p
,
q
such that
p
3
−
q
7
=
p
−
q
p ^3-q^7=p-q
p
3
−
q
7
=
p
−
q
.
1
1
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is (1+a^2)(1+b^2)(1+c^2) perfect square when ab + bc + ca = 1 ??
Integers
a
,
b
a, b
a
,
b
and
c
c
c
satisfy the condition
a
b
+
b
c
+
c
a
=
1
ab + bc + ca = 1
ab
+
b
c
+
c
a
=
1
. Is it true that the number
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
(1+a^2)(1+b^2)(1+c^2)
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
is a perfect square? Why?
4
1
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orthocenter of APQ lies on MN, cyclic ABCD, AB = AD, MN = BN + DM
In the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
A
D
AB = AD
A
B
=
A
D
, points
M
M
M
and
N
N
N
lie on the sides
C
D
CD
C
D
and
B
C
BC
BC
respectively so that
M
N
=
B
N
+
D
M
MN = BN + DM
MN
=
BN
+
D
M
. Lines
A
M
AM
A
M
and
A
N
AN
A
N
meet the circumcircle of
A
B
C
D
ABCD
A
BC
D
again at points
P
P
P
and
Q
Q
Q
respectively. Prove that the orthocenter of the triangle
A
P
Q
APQ
A
PQ
lies on the segment
M
N
MN
MN
.