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Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
1996 Japan MO Finals
1996 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
5
1
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Changing powers of 2 in binary representation to q-powers
Let
q
q
q
be a real number with
1
+
5
2
<
q
<
2
\frac{1+\sqrt{5}}{2}<q<2
2
1
+
5
<
q
<
2
. If a positive integer
n
n
n
is represented in binary system as
2
k
+
a
k
−
1
2
k
−
1
+
⋯
+
2
a
1
+
a
0
2^k+a_{k-1}2^{k-1}+\cdots +2a_1+a_0
2
k
+
a
k
−
1
2
k
−
1
+
⋯
+
2
a
1
+
a
0
, where
a
i
∈
{
0
,
1
}
a_i\in\{0,1\}
a
i
∈
{
0
,
1
}
, define
p
n
=
q
k
+
a
k
−
1
q
k
−
1
+
⋯
+
q
a
1
+
a
0
.
p_n=q^k+a_{k-1}q^{k-1}+\cdots +qa_1+a_0.
p
n
=
q
k
+
a
k
−
1
q
k
−
1
+
⋯
+
q
a
1
+
a
0
.
Prove that there exist infinitely many positive integers
k
k
k
with the property that there is no
l
∈
N
l\in\mathbb{N}
l
∈
N
such that
p
2
k
<
p
l
<
p
2
k
+
1
p_{2k}<p_l< p_{2k+1}
p
2
k
<
p
l
<
p
2
k
+
1
.
4
1
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Maximum of minimum angle θ in regular tetradhedron
Let
θ
\theta
θ
be the maximum of the six angles between six edges of a regular tetrahedron in space and a fixed plane. When the tetrahedron is rotated in space, find the maximum of
θ
\theta
θ
.
3
1
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Sequence from non-integer x is not periodic
Let
x
>
1
x>1
x
>
1
be a real number which is not an integer. For each
n
∈
N
n\in\mathbb{N}
n
∈
N
, let
a
n
=
⌊
x
n
+
1
⌋
−
x
⌊
x
n
⌋
a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor
a
n
=
⌊
x
n
+
1
⌋
−
x
⌊
x
n
⌋
. Prove that the sequence
(
a
n
)
(a_n)
(
a
n
)
is not periodic.
2
1
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Find gcd(5^m+7^m,5^n+7^n) for coprime m,n
Let
m
,
n
m,n
m
,
n
be positive integers with
(
m
,
n
)
=
1
(m,n)=1
(
m
,
n
)
=
1
. Find
(
5
m
+
7
m
,
5
n
+
7
n
)
(5^m+7^m,5^n+7^n)
(
5
m
+
7
m
,
5
n
+
7
n
)
.
1
1
Hide problems
Every angle in triangle σ is greater than θ in partition
A plane is partitioned into triangles. Let
T
0
\mathcal{T}_0
T
0
denote the set of vertices of triangles in the partition. Let
A
B
C
ABC
A
BC
be a triangle with
A
,
B
,
C
∈
T
0
A,B,C\in\mathcal{T}_0
A
,
B
,
C
∈
T
0
and
θ
\theta
θ
be its smallest angle. Assume that no point of
T
0
\mathcal{T}_0
T
0
lies inside the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that there exists a triangle
σ
\sigma
σ
in the partition such that its intersection with
△
A
B
C
\triangle ABC
△
A
BC
is nonempty and whose every angle is greater than
θ
\theta
θ
.