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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2011 Iran MO (3rd Round)
2011 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(8)
8
1
Hide problems
covering sequences
We call the sequence
d
1
,
.
.
.
.
,
d
n
d_1,....,d_n
d
1
,
....
,
d
n
of natural numbers, not necessarily distinct, covering if there exists arithmetic progressions like
c
1
+
k
d
1
c_1+kd_1
c
1
+
k
d
1
,....,
c
n
+
k
d
n
c_n+kd_n
c
n
+
k
d
n
such that every natural number has come in at least one of them. We call this sequence short if we can not delete any of the
d
1
,
.
.
.
.
,
d
n
d_1,....,d_n
d
1
,
....
,
d
n
such that the resulting sequence be still covering. a) Suppose that
d
1
,
.
.
.
.
,
d
n
d_1,....,d_n
d
1
,
....
,
d
n
is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions
a
1
+
k
d
1
,
.
.
.
.
.
,
a
n
+
k
d
n
a_1+kd_1,.....,a_n+kd_n
a
1
+
k
d
1
,
.....
,
a
n
+
k
d
n
, and suppose that
p
p
p
is a prime number that
p
p
p
divides
d
1
,
.
.
.
.
,
d
k
d_1,....,d_k
d
1
,
....
,
d
k
but it does not divide
d
k
+
1
,
.
.
.
.
,
d
n
d_{k+1},....,d_n
d
k
+
1
,
....
,
d
n
. Prove that the remainders of
a
1
,
.
.
.
.
,
a
k
a_1,....,a_k
a
1
,
....
,
a
k
modulo
p
p
p
contains all the numbers
0
,
1
,
.
.
.
.
.
,
p
−
1
0,1,.....,p-1
0
,
1
,
.....
,
p
−
1
. b) Write anything you can about covering sequences and short covering sequences in the case that each of
d
1
,
.
.
.
.
,
d
n
d_1,....,d_n
d
1
,
....
,
d
n
has only one prime divisor.proposed by Ali Khezeli
7
1
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predicting function
Suppose that
f
:
P
(
N
)
⟶
N
f:P(\mathbb N)\longrightarrow \mathbb N
f
:
P
(
N
)
⟶
N
and
A
A
A
is a subset of
N
\mathbb N
N
. We call
f
f
f
A
A
A
-predicting if the set
{
x
∈
N
∣
x
∉
A
,
f
(
A
∪
x
)
≠
x
}
\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}
{
x
∈
N
∣
x
∈
/
A
,
f
(
A
∪
x
)
=
x
}
is finite. Prove that there exists a function that for every subset
A
A
A
of natural numbers, it's
A
A
A
-predicting.proposed by Sepehr Ghazi-Nezami
6
3
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bacteria filling cells of a table
Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an
m
×
n
m\times n
m
×
n
table such that no two of them overlap?proposed by Mahyar Sefidgaran
how many functions are there?
a
a
a
is an integer and
p
p
p
is a prime number and we have
p
≥
17
p\ge 17
p
≥
17
. Suppose that
S
=
{
1
,
2
,
.
.
.
.
,
p
−
1
}
S=\{1,2,....,p-1\}
S
=
{
1
,
2
,
....
,
p
−
1
}
and
T
=
{
y
∣
1
≤
y
≤
p
−
1
,
o
r
d
p
(
y
)
<
p
−
1
}
T=\{y|1\le y\le p-1,ord_p(y)<p-1\}
T
=
{
y
∣1
≤
y
≤
p
−
1
,
or
d
p
(
y
)
<
p
−
1
}
. Prove that there are at least
4
(
p
−
3
)
(
p
−
1
)
p
−
4
4(p-3)(p-1)^{p-4}
4
(
p
−
3
)
(
p
−
1
)
p
−
4
functions
f
:
S
⟶
S
f:S\longrightarrow S
f
:
S
⟶
S
satisfying
∑
x
∈
T
x
f
(
x
)
≡
a
\sum_{x\in T} x^{f(x)}\equiv a
∑
x
∈
T
x
f
(
x
)
≡
a
(
m
o
d
(mod
(
m
o
d
p
)
p)
p
)
. proposed by Mahyar Sefidgaran
fighting circles
We call two circles in the space fighting if they are intersected or they are clipsed. Find a good necessary and sufficient condition for four distinct points
A
,
B
,
A
′
,
B
′
A,B,A',B'
A
,
B
,
A
′
,
B
′
such that each circle passing through
A
,
B
A,B
A
,
B
and each circle passing through
A
′
,
B
′
A',B'
A
′
,
B
′
are fighting circles.proposed by Ali Khezeli
5
5
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4
5
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3
5
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2
6
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1
6
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