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Problems
Contests
National and Regional Contests
Israel Contests
Israel National Olympiad
2015 Israel National Olympiad
2015 Israel National Olympiad
Part of
Israel National Olympiad
Subcontests
(7)
7
1
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Israel 2015 Q7 - Fibonacci prime power divisibility
The Fibonacci sequence
F
n
F_n
F
n
is defined by
F
0
=
0
,
F
1
=
1
F_0=0,F_1=1
F
0
=
0
,
F
1
=
1
and the recurrence relation
F
n
=
F
n
−
1
+
F
n
−
2
F_n=F_{n-1}+F_{n-2}
F
n
=
F
n
−
1
+
F
n
−
2
for all integers
n
≥
2
n\geq2
n
≥
2
. Let
p
≥
3
p\geq3
p
≥
3
be a prime number. [*] Prove that
F
p
−
1
+
F
p
+
1
−
1
F_{p-1}+F_{p+1}-1
F
p
−
1
+
F
p
+
1
−
1
is divisible by
p
p
p
. [*] Prove that
F
p
k
+
1
−
1
+
F
p
k
+
1
+
1
−
(
F
p
k
−
1
+
F
p
k
+
1
)
F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)
F
p
k
+
1
−
1
+
F
p
k
+
1
+
1
−
(
F
p
k
−
1
+
F
p
k
+
1
)
is divisible by
p
k
+
1
p^{k+1}
p
k
+
1
for any positive integer
k
k
k
.
6
1
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Israel 2015 Q6 - Shifting lamps process
Let
n
≥
1
n\geq1
n
≥
1
be a positive integer.
n
n
n
lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp is on at minute
t
+
1
t+1
t
+
1
if and only if at minute
t
t
t
, exactly one of its neighbors is on (the two lamps at the ends have one neighbor each, all other lamps have two neighbors).For which values of
n
n
n
can we guarantee that all lamps will be off after some time?
5
1
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Israel 2015 Q5 - Tetrahedron and inscribed spheres
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron. Denote by
S
1
S_1
S
1
the inscribed sphere inside it, which is tangent to all four faces. Denote by
S
2
S_2
S
2
the outer escribed sphere outside
A
B
C
ABC
A
BC
, tangent to face
A
B
C
ABC
A
BC
and to the planes containing faces
A
B
D
,
A
C
D
,
B
C
D
ABD,ACD,BCD
A
B
D
,
A
C
D
,
BC
D
. Let
K
K
K
be the tangency point of
S
1
S_1
S
1
to the face
A
B
C
ABC
A
BC
, and let
L
L
L
be the tangency point of
S
2
S_2
S
2
to the face
A
B
C
ABC
A
BC
. Let
T
T
T
be the foot of the perpendicular from
D
D
D
to the face
A
B
C
ABC
A
BC
.Prove that
L
,
T
,
K
L,T,K
L
,
T
,
K
lie on one line.
4
1
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Israel 2015 Q4 - Divisibility conditions
Let
k
,
m
,
n
k,m,n
k
,
m
,
n
be positive integers such that
n
m
n^m
n
m
is divisible by
m
n
m^n
m
n
, and
m
k
m^k
m
k
is divisible by
k
m
k^m
k
m
.[*] Prove that
n
k
n^k
n
k
is divisible by
k
n
k^n
k
n
. [*] Find an example of
k
,
m
,
n
k,m,n
k
,
m
,
n
satisfying the above conditions, where all three numbers are distinct and bigger than 1.
3
1
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Israel 2015 Q3 - Root expression is an integer
Prove that the number
(
76
1
77
3
−
75
3
−
5775
3
+
1
76
77
3
+
75
3
+
5775
3
)
3
\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3
(
3
77
−
3
75
1
−
3
5775
76
+
3
77
+
3
75
76
+
3
5775
1
)
3
is an integer.
2
1
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Israel 2015 Q2 - Triangle area from altitudes
A triangle is given whose altitudes' lengths are
1
5
,
1
5
,
1
8
\frac{1}{5},\frac{1}{5},\frac{1}{8}
5
1
,
5
1
,
8
1
. Evaluate the triangle's area.
1
1
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Israel 2015 Q1 - Timely diophantine equation
[*] Find an example of three positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
31
a
+
30
b
+
28
c
=
365
31a+30b+28c=365
31
a
+
30
b
+
28
c
=
365
. [*] Prove that any triplet
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying the above condition, also satisfies
a
+
b
+
c
=
12
a+b+c=12
a
+
b
+
c
=
12
.